From the question it is given that, Sum of the terms Sn = \[\mathbf{3069}/\mathbf{512}\] First term a = \[3\] Common ratio r \[=\text{ }\left( 3/2 \right)/3\] \[\begin{array}{*{35}{l}} =\text{...
How many terms of the G.P.
If the
terms of a G.P. are x, y and z respectively, prove that x, y and z are in G.P.
From the question it is given that, \[\begin{array}{*{35}{l}} {{a}_{4}}~=\text{ }x \\ {{a}_{10}}~=\text{ }y \\ {{a}_{16}}~=\text{ }z \\ \end{array}\] Now, we have to show that x, y and z are in...
The fourth term of a G.P. is the square of its second term and the first term is
. Find its 7th term.
From the question it is given that, The fourth term of a G.P. is the square of its second term \[=\text{ }{{a}_{4}}~=\text{}{{({{a}_{2}})}^{2}}\] The first term \[{{a}_{1}}~=\text{ }\text{ }3\] We...
Find the geometric progression whose 4th term is
and 7th term is
.
From the question it is given that, The geometric progression whose 4th term \[{{a}_{4}}~=\text{ }54\] The geometric progression whose 7th term \[{{a}_{7}}~=\text{ }1458\] We know that, an = arn – 1...
The sum of first
terms of an A.P. is
and its first term is
. Find its 25th term.
From the question it is given that, First term a = \[10\] The sum of first 14 terms of an A.P. = \[1505\] 25th term = ? We know that, \[{{S}_{n}}~=\text{ }\left( n/2 \right)\text{ }\left[ 2a\text{...
If the third term of an A.P. is
and the ratio of its 6th term to the 10th term is
, thenfind the sum of first
terms of this A.P.
From the question it is given that, The third term of an A.P. a3 = \[5\] The ratio of its 6th term to the 10th term \[~{{a}_{6}}~:\text{ }{{a}_{10}}~=\text{ }7\text{ }:\text{ }13\] We know that,...
(i) How many terms of the A.P.
make the sum
? (ii) Solve the equation
From the question it is given that, Terms of the A.P. is \[-6,\text{ }\left( -11/2 \right)\text{ }\text{ }5,\text{ }\ldots \] The first term a = \[-6\] Common difference \[\begin{array}{*{35}{l}}...
Find the sum :
From the question it is given that, First term a = \[18\] Common difference d = \[15{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\text{ }18\] \[\begin{array}{*{35}{l}} =\text{ }31/2\text{ }\text{ }18 \\...
Find the sum of first
terms of an A.P. whose nth term is
.
From the question it is given that, nth term is \[15\text{ }\text{ }4n\] So, \[{{a}_{n}}~=\text{ }15\text{ }\text{ }4n\] Now, we start giving values, \[1,\text{ }2,\text{ }3,\text{ }\ldots \] in the...
The angles of a quadrilateral are in A.P. If the greatest angle is double of the smallest angle, find all the four angles.
From the question it is given that, The angles of a quadrilateral are in A.P. Greatest angle is double of the smallest angle Let us assume the greatest angle of the quadrilateral is a + 3d, Then,...
The sum of three numbers in A.P. is
and the product is 8. Find the numbers.
From the question it is given that, The sum of three numbers in A.P. = \[-3\] The product of three numbers in A.P. = \[8\] Let us assume the 3 numbers which are in A.P. are, a – d, a, a + d Now...
How many three digit numbers are divisible by
?
The three digits numbers which are divisible by \[9\] are \[108,\text{ }117,\text{ }126,\text{ }\ldots ,\text{ }999\] Then, first term a = \[108\] Common difference = \[9\] Last term = \[999\] We...
Which term of the list of numbers
is the first negative term?
From the question it is given that, First term a = \[20\] Common difference d = \[19{\scriptscriptstyle 1\!/\!{ }_4}\text{ }\text{ }20\text{ }=\text{ }77/4\text{ }\text{ }20\text{ }=\text{ }\left(...
The
term of an A.P. is twice its
term. Show that its
term is four times its
term.
From the question it is given that, The \[{{24}^{th}}\] term of an A.P. is twice its \[{{10}^{th}}\] term = \[{{a}_{24}}~=\text{ }2{{a}_{10}}\] We have to show that, \[{{72}^{nd}}\] term is four...
Which term of the list of numbers
?
From the question it is given that, First term a = \[5\] nth term = \[-55\] Common difference d = \[2\text{ }\text{ }5\text{ }=\text{ }\text{ }3\] We know that, an = a + (n – 1)d...
If the
and the
terms of an A.P. are
and
, respectively, then which term of this A.P. is zero?
From the question it is given that, \[\begin{array}{*{35}{l}} {{a}_{3}}~=\text{ }4 \\ {{a}_{9}}~=\text{ }\text{ }8 \\\end{array}\] We know that, \[{{a}_{3}}~=\text{ }a\text{ }+\text{ }2d\text{...
The
term of an A.P. is equal to three times its
term. If its
term is
, find the A.P.
From the question it is given that, \[{{a}_{19}}~=\text{ }{{19}^{th}}~\] term of an A.P. is equal to three times its \[{{6}^{th}}\] term = \[3{{a}_{6}}\] \[{{a}_{9}}~=\text{ }19\] As we know, an = a...
The
term of an A.P. is 5 more than twice its
term. If the
term of the A.P. is
, then find the
term.
From the question it is given that, \[{{a}_{17}}~=\text{ }5\] more than twice its \[{{8}^{th}}\] term = \[2{{a}_{8}}~+\text{ }5\] \[\begin{array}{*{35}{l}} {{a}_{11}}~=\text{ }43 \\...
If the
term of an A.P. is
and the
term is
more than its
term, then find the A.P.
From the question it is given that, \[{{a}_{8}}~=\text{ }31\] \[{{a}_{15}}\] = the \[{{15}^{th}}\] term is \[16\] more than its \[{{11}^{th}}\] term = \[{{a}_{11}}~+\text{ }16\] we know that, an = a...
Find the 6th term from the end of the A.P.
From the question it is given that, First term a = \[17\] Common difference = \[14\text{ }\text{ }17\text{ }=\text{ }\text{ }3\] Last term l = \[-40\] \[\begin{array}{*{35}{l}} L\text{ }=\text{...
Show that the list of numbers
form an A.P. Find its
term and the
.
From the question, The first term a = \[9\] Then, difference d = \[12\text{ }\text{ }9\text{ }=\text{ }3\] \[\begin{array}{*{35}{l}} 15\text{ }\text{ }12\text{ }=\text{ }3 \\ 18\text{ }\text{...
The nth term of an A.P. is
. Find the common difference.
From the question it is given that, nth term is \[\mathbf{6n}\text{ }+\text{ }\mathbf{2}\] So, \[{{T}_{n}}~=\text{ }6n\text{ }+\text{ }2\] Now, we start giving values, \[1,\text{ }2,\text{ }3,\text{...
Verify that each of the following lists of numbers is an A.P., and the write its next three terms: (i)
(ii)
From the question it is given that, First term a = 0 Common difference \[=\text{ }{\scriptscriptstyle 1\!/\!{ }_4}\text{ }\text{ }0\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_4}\] So, next three...
Write the first four terms of the A.P. when its first term is
and the common difference is
.
From the question it is given that, First term a = \[-5\] Common difference d = \[-3\] Then the first four terms are = \[\text{ }5\text{ }+\text{ }\left( -3 \right)\text{ }=\text{ }-5\text{ }\text{...
If the first term of a G.P. is
and the sum of first three terms is
, find the common ratio.
From the question it is given that, First term of a G.P. is a = \[5\] The sum of first three terms is S3 = \[31/5\] We know that, \[{{S}_{n}}~=\text{ }a({{r}^{n}}~\text{ }1)/\left( r\text{ }\text{...
Find the third term of a G.P. whose common ratio is
and the sum of whose first seven terms is
.
From the question it is given that, Common ratio r = \[3\] \[{{S}_{7}}~=\text{ }2186\] We know that, \[{{S}_{n}}~=\text{ }a({{r}^{n}}~\text{ }1)/\left( r\text{ }\text{ }1 \right)\]...
In a G.P. the first term is
, the last term is
, and the sum is
. Find the common ratio.
From the question it is given that, First term a is = \[7\] Then, last term is = \[448\] Sum = \[889\] We know that, last term = \[~a{{r}^{n\text{ }\text{ }1}}\] \[\begin{array}{*{35}{l}}...
Find the first term of the G.P. whose common ratio is
, last term is
and the sum of whose terms is
.
From the question it is given that, Common ratio r = \[3\] Last term = \[486\] Sum of the terms = \[728\] We know that, \[{{S}_{n}}~=\text{ }a({{r}^{n}}~\text{ }1)/\left( r\text{ }\text{ }1...
The first term of G.P. is
and
term is
. Find the sum of its first
terms.
From the question it is given that, First term a = \[27\] \[{{8}^{th}}~term\text{ }{{a}_{8}}~=\text{ }1/81\] Then, \[\begin{array}{*{35}{l}} {{a}_{n}}~=\text{ }a{{r}^{n\text{ }\text{ }1}} \\...
The
and
terms of a geometric series are
and sum
, respectively. Find the sum of the series up to
terms.
From the question it is given that, \[\begin{array}{*{35}{l}} {{a}_{2}}~=\text{ }-{\scriptscriptstyle 1\!/\!{ }_2} \\ {{a}_{5}}~=\text{ }1/16 \\ \end{array}\] We know that, \[{{a}_{2}}~=\text{...
How many terms of the
will make the sum
?
From the question it is given that, Terms of G.P. is \[\mathbf{2}/\mathbf{9}\text{ }\text{ }\mathbf{1}/\mathbf{3}\text{ }+\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }+\text{ }\ldots \] Sum of...
(i) How many terms of the G.P.
are needed to give the sum
? (ii) How many terms of the G.P.
must be taken to have their sum equal to
?
From the question it is given that, Terms of the G.P. \[\mathbf{3},\text{ }{{\mathbf{3}}^{\mathbf{2}}},\text{ }{{\mathbf{3}}^{\mathbf{3}}},\text{ }\ldots \] Sum of the terms = \[120\] The first term...
The
term of a G.P. is
and the sum of its n terms is
. If its common ratio is
, then find its first term.
From the question it is given that, The nth term of a G.P. \[{{T}_{n}}~=\text{ }128\] The sum of its n terms \[{{S}_{n}}~=\text{ }255\] Common ratio r = \[2\] We know that, \[{{T}_{n}}~=\text{...
Find the sum of the series
From the question it is given that, First term a = \[81\] \[\begin{array}{*{35}{l}} r\text{ }=\text{ }-27/81 \\ =\text{ }-1/3 \\ \end{array}\] Last term l =\[~-1/27\] \[\begin{array}{*{35}{l}} ...
Find the sum of: (iii)
terms of the GP
(iv)
terms and n terms of the series
From the question, First term a = \[1\], Common ratio \[r\text{ }=\text{ }-2/3\text{ }\times \text{ }1=\text{ }-2/3\] Number of terms n = \[6\] So, \[\begin{array}{*{35}{l}} {{S}_{6}}~=\text{...
Find the sum of: (i)
terms of the series
(ii)
terms of series
From the question, First term a = \[2\], Common ratio r = \[6/2\text{ }=\text{ }3\] Number of terms n = \[20\] So, \[\begin{array}{*{35}{l}} {{S}_{20}}~=\text{ }a({{r}^{n}}~\text{ }1)/r\text{...
Three numbers are in A.P. and their sum is
. If
and
are added to these numbers respectively, the resulting numbers are in G.P. Find the numbers.
From the question it is give that, The sum of first three terms of a A.P. is \[15\] Let us assume three numbers are a – d, a, a + d. The sum of three terms = \[a\text{ }\text{ }d\text{ }+\text{...
The sum of first three terms of a G.P. is
and their product is
. Find the common ratio and the terms.
From the question it is give that, The sum of first three terms of a G.P. is \[\mathbf{39}/\mathbf{10}\] The product of first three terms of a G.P. is \[1\] Let us assume that a be the first term...
Find the geometric progression whose
term is
and the
term is
.
From the question it is given that, \[{{4}^{th}}\] term \[{{a}_{4}}~=\text{ }54\] \[{{7}^{th}}\] term \[{{a}_{7}}~=\text{ }1458\] \[\begin{array}{*{35}{l}} a{{r}^{3}}~=\text{ }54 \\...
If a,
and
are in G.P., then find the values(s) of a.
From the question, \[\begin{array}{*{35}{l}} {{({{a}^{2}}~+\text{ }2)}^{2}}~=\text{ }a({{a}^{3}}~+\text{ }10) \\ {{a}^{4}}~+\text{ }4\text{ }=\text{ }{{a}^{4}}~+\text{ }10a \\ 4{{a}^{2}}~\text{...
The
terms of a G.P. are p, q and s, respectively. Show that
.
From the question it is given that, \[\begin{array}{*{35}{l}} {{a}_{5}}~=\text{ }p \\ {{a}_{8}}~=\text{ }q \\ {{a}_{11}}~=\text{ }s \\ \end{array}\] Now we have to prove that,...
If the fourth, seventh and tenth terms of a G.P. are x, y, z respectively, prove that x, y, z are in G.P.
From the question it is given that, a4 = x a7 = y a10 = z Now we have to prove that, x, y, z are in G.P. Then, by the formula \[{{a}_{n}}~=\text{ }a{{r}^{n\text{ }\text{ }1}}\]...
Find the value of x such that, (iii) x,
are first three terms of a G.P. Find the value of x.
From the question, \[\begin{array}{*{35}{l}} {{\left( x\text{ }+\text{ }3 \right)}^{2}}~=\text{ }x\left( x\text{ }+\text{ }9 \right) \\ {{x}^{2}}~+\text{ }6x\text{ }+\text{ }9\text{ }=\text{...
Find the value of x such that, (i)
are three consecutive terms of a G.P. (ii)
are three consecutive terms of a G.P.
From the question, \[\begin{array}{*{35}{l}} {{x}^{2}}~=\text{ }-2/7\text{ }\times \text{ }-7/2 \\ {{x}^{2}}~=\text{ }1 \\ x\text{ }=\text{ }\pm \text{ }1 \\ \end{array}\] Therefore, \[x\text{...
Find the number of terms of a G.P. whose first term is
, common ratio is
and the last term is
.
From the question it is given that, First term of G.P. a = \[{\scriptscriptstyle 3\!/\!{ }_4}\] Common ratio (r) = \[2\] Last term = \[384\] Then, by the formula \[{{a}_{n}}~=\text{ }a{{r}^{n\text{...
Determine the
term of a G.P. whose
term is
and common ratio is
.
From the question it is given that, a8 = \[192\] and r = \[2\] Then, by the formula \[{{a}_{n}}~=\text{ }a{{r}^{n\text{ }\text{ }1}}\] \[\begin{array}{*{35}{l}} {{a}_{8}}~=\text{ }a{{r}^{8\text{...
Which term of the G.P. (i)
(ii)
From the question it is given that, Last term = 128 First term a = 2, Then, \[\begin{array}{*{35}{l}} r\text{ }=\text{ }\left( 2\surd 2 \right)\text{ }\div \text{ }\left( 2 \right) \\ r\text{...
(vii) Find the 6th term from the end of the list of numbers
.
From the question it is given that, Last term = 12288 First term a = 3, Then, \[\begin{array}{*{35}{l}} r\text{ }=\text{ }\left( -6 \right)\text{ }\div \text{ }\left( 3 \right) \\ r\text{ }=\text{...
(v) Find the 10th and nth terms of the list of numbers
(vi) Find the 6th and the nth terms of the list of numbers
From the question it is given that, First term a = 5, Then, \[\begin{array}{*{35}{l}} ~r\text{ }=\text{ }\left( 25 \right)\text{ }\div \text{ }\left( 5 \right) \\ r\text{ }=\text{ }\left( 25...
(iii) Find the 15th term of the series
(iv) Find the nth term of the list of numbers
From the question, First term \[a\text{ }=\text{ }\surd 3\] Then, \[\begin{array}{*{35}{l}} r\text{ }=\text{ }\left( 1/\surd 3 \right)\text{ }\div \text{ }\left( \surd 3 \right) \\ r\text{ }=\text{...
1. (i) Find the next term of the list of numbers
(ii) Find the next term of the list of numbers
From the question, First term a = \[1/6\] Then, \[\begin{array}{*{35}{l}} ~r\text{ }=\text{ }\left( 1/3 \right)\text{ }\div \text{ }\left( 1/6 \right) \\ r\text{ }=\text{ }\left( 1/3 \right)\text{...
(iii) Find the sum of all multiples of 9 lying between
and
. (iv) Find the sum of all natural numbers less than
which are divisible by
.
(iii) The multiples of 9 lying between \[300\] and \[700\] are: \[306,\text{ }315,\text{ }324,\text{ }\ldots \ldots \] This form an A.P. where a = 306 and d = 9 The last term in this series is...
(i) Find the sum of all two digit natural numbers which are divisible by
. (ii) Find the sum of all natural numbers between
and
which are divisible by
.
(i) The two-digit natural numbers which are divisible by \[4\] are: \[4,\text{ }8,\text{ }12,\text{ }16,\text{ }\ldots ..\] This form an A.P. The last term in this series is found out by dividing...
(i) Find the sum of first
positive integers. (ii) Find the sum of first
multiples of
.
(i) First \[1000\] positive integers are: \[1,\text{ }2,\text{ }3,\text{ }4,\text{ }\ldots \ldots ..,\text{ }1000\] This is an A.P with first term a = \[1\] and common difference d = \[1\] We know...
If Sn denotes the sum of first n terms of an A.P., prove that
.
We know that, \[{{S}_{n}}~=\text{ }n/2\text{ }\times \text{ }\left[ 2a\text{ }+\text{ }\left( n\text{ }\text{ }1 \right)d \right]\] So, \[\begin{array}{*{35}{l}} {{S}_{10}}~=\text{ }10/2\text{...
In an A.P., the sum of its first n terms is
. Find its
term.
Given, \[{{S}_{n}}~=\text{ }6n\text{ }\text{ }{{n}^{2}}\] Now, \[{{S}_{1}}~=\text{ }6\left( 1 \right)\text{ }\text{ }{{\left( 1 \right)}^{2}}~=\text{ }6\text{ }\text{ }1\text{ }=\text{ }5\] So, the...
The sum of first six terms of an arithmetic progression is
. The ratio of the
term to the
term is
. Calculate the first and the thirteenth term.
Given, \[{{S}_{6}}~=\text{ }42\text{ }and\text{ }{{T}_{10}}/{{T}_{30}}~=\text{ }1/3\] We know that, \[\begin{array}{*{35}{l}} {{S}_{n}}~=\text{ }\left( n/2 \right)\text{ }\left( 2a\text{ }+\text{...
Show that
form an A.P. where an is defined as
. Also find the sum of first
terms.
From the question it is given that, nth term is \[3+4n\] So, \[{{\mathbf{a}}_{\mathbf{n}}}~=\text{ }\mathbf{3}\text{ }+\text{ }\mathbf{4n}\] Now, we start giving values, \[1,2,3.....\] in the place...
If the sum of first
terms of an A.P. is
and that of the first
terms is
, find the sum of first 10 terms.
From the question it is given that, \[\begin{array}{*{35}{l}} {{S}_{6}}~=\text{ }36 \\ {{S}_{16}}~=\text{ }256 \\ \end{array}\] We know that, \[\begin{array}{*{35}{l}} {{S}_{n}}~=\text{ }\left(...
(i) Find the sum of first
terms of the A.P. whose second and third terms are
and
, respectively. (ii) The
term of A.P is
and
term is
. Find the first term and the common difference. Hence, find the sum of first 8 term of the A.P.
From the question it is given that, \[{{T}_{2}}~=\text{ }14,\text{ }{{T}_{3}}~=\text{ }18\] So, common difference d = \[{{T}_{3}}~\text{ }{{T}_{2}}\] \[\begin{array}{*{35}{l}} =\text{ }18\text{...
In an A.P., the fourth and sixth terms are
and
, respectively. Find the: (iii) sum of the first
terms.
From the question it is given that, \[{{T}_{4}}~=\text{ }8\text{ }and\text{ }{{T}_{6}}~=\text{ }14\] ⇒ \[a\text{ }+\text{ }3d\text{ }=\text{ }8\]… (i) ⇒ \[a\text{ }+\text{ }5d\text{ }=\text{ }14\]…...
In an A.P., the fourth and sixth terms are
and
, respectively. Find the: (i) first term (ii) common difference
From the question it is given that, \[{{T}_{4}}~=\text{ }8\text{ }and\text{ }{{T}_{6}}~=\text{ }14\] ⇒ \[a\text{ }+\text{ }3d\text{ }=\text{ }8\]… (i) ⇒ \[a\text{ }+\text{ }5d\text{ }=\text{ }14\]…...
Find the sum of first
terms, of an A.P. in which d =
and
is
.
From the question it is given that, Common difference d = \[7\] \[\begin{array}{*{35}{l}} {{a}_{22}}~=\text{ }149 \\ n\text{ }=\text{ }22 \\ \end{array}\] we know that, \[\begin{array}{*{35}{l}}...
(i) How many terms of the A.P.
are needed to give the sum
? Also find the last term. (ii) How many terms of the A.P.
must be taken so that the sum is
? Explain the double answer.
From the question it is given that, First term a = \[25\] Common difference d = \[22\text{ }\text{ }25\text{ }=\text{ }\text{ }3\] Sum = \[116\] \[\begin{array}{*{35}{l}} {{S}_{n}}~=\text{ }\left(...
Solve for
.
From the question, First term a = \[1\] Difference d = \[4\text{ }\text{ }1\text{ }=\text{ }3\] n = x \[\begin{array}{*{35}{l}} x\text{ }=\text{ }a\text{ }=\text{ }\left( n\text{ }\text{ }1...
The first and the last terms of an A.P. are
and
, respectively. If the common difference is
, how many terms are there and what is their sum?
From the question it is give that, First term a = \[17\] Last term (l) = \[350\] Common difference d = \[9\] We know that, l = \[{{T}_{n}}~=\text{ }a\text{ }+\text{ }\left( n\text{ }\text{ }1...
(i) The first term of an A.P. is 5, the last term is
and the sum is
. Find the number of terms and the common difference. (ii) The sum of first
terms of an A.P. is
and its first term is
. Find its
th term.
From the question it is give that, First term a = \[5\] Last term = \[45\] Then, sum = \[400\] We know that, last term = a + (n – 1)d \[\begin{array}{*{35}{l}} 45\text{ }=\text{ }5\text{ }+\text{...
In an A.P. (with usual notations) : (v) given
, find d.
From the question it is given that, First term a = \[3\] n = \[8\] S = \[192\] We know that, \[{{S}_{n}}\] = (n/2) (2a + (n – 1)d) Therefore, common difference d is \[6\].
In an A.P. (with usual notations) : (iii) given d =
,
, find a and a9. (iv) given
, find n and d
From the question it is given that, Common difference d = \[5\] \[{{\mathbf{S}}_{\mathbf{9}}}~=\text{ }\mathbf{75}\] We know that, an = a + (n – 1)d \[\begin{array}{*{35}{l}} {{a}_{9}}~=\text{...
In an A.P. (with usual notations) : (i) given
(ii) given
, find d and
From the question, First term a = \[5\] Then common difference d = \[3\] \[{{a}_{n}}~=\text{ }50\], We know that, \[{{a}_{n}}~=\text{ }a\text{ }+\text{ }\left( n\text{ }\text{ }1 \right)d\]...
Find the sums given below : (i)
(ii)
From the question, First term a = \[34\], Difference d = \[32\text{ }\text{ }34\text{ }=\text{ }-2\] So, common difference d = \[-2\] Last term Tn = 10 We know that, \[{{T}_{n}}\] = a + (n – 1)d...
1. Find the sum of the following A.P.s : (i)
terms (ii)
terms
From the question, First term a = \[2\] Then, d = \[7\text{ }\text{ }2\text{ }=\text{ }5\] \[12\text{ }\text{ }7\text{ }=\text{ }5\] So, common difference d = \[5\] \[\begin{array}{*{35}{l}} ...
The sum of the first three terms of an A.P.is
. If the product of the first and the third terms exceeds the second term by
, find the A.P.
From the question it is given that, sum of the first three terms of an A.P. is \[33\]. Let us assume the \[3\] numbers which are in A.P. are, a – d, a, a + d Now adding \[3\] numbers = a – d + a + a...
The sum of three numbers in A.P. is
and the ratio of first number to the third number is
. Find the numbers.
From the question it is given that, sum of three numbers in A.P. = \[30\] The ratio of first number to the third number is \[3:7\] Let us assume the 3 numbers which are in A.P. are, a – d, a, a + d...
The sum of three numbers in A.P. is
and their product is
. Find the numbers.
From the question it is given that, The sum of three numbers in A.P. = \[3\] Given, Their product = \[-35\] Let us assume the \[3\] numbers which are in A.P. are, a – d, a, a + d Now adding \[3\]...
If the numbers
are in A.P., find the value of n.
From the question it is given that, \[\mathbf{n}\text{ }\text{ }\mathbf{2},\text{ }\mathbf{4n}\text{ }\text{ }\mathbf{1}\text{ }\mathbf{and}\text{ }\mathbf{5n}\text{ }+\text{ }\mathbf{2}\] are in...
(i) How many two digit numbers are divisible by
? (ii) Find the number of natural numbers between
and
which are divisible by both
and
.
The two digits numbers divisible by \[3\] are, \[12,\text{ }15,\text{ }18,\text{ }21,\text{ }24,\ldots ..,99\]. The above numbers are A.P. So, first number a = \[12\] Common difference d =...
Which term of the A.P.
will be
more than its
term?
From the question it is given that, First term a = \[3\] Common difference d = \[10\text{ }\text{ }3\text{ }=\text{ }7\] Then, \[\begin{array}{*{35}{l}} {{T}_{13}}~=\text{ }a\text{ }+\text{ }12d \\...
If
term of an A.P. is zero, prove that its
term is triple of its
term.
Froom the question it is given that, \[{{T}_{8}}~=\text{ }0\] We have to prove that, \[{{38}^{th}}\] term is triple of its \[{{18}^{th}}\] term = \[{{T}_{38}}~=\text{ }3{{T}_{18}}\]...
(i) The
term of an A.P. is
more than twice its
term. If the
term of the A.P. is
, find its nth term. (ii) The sum of
and
terms of an A.P. is
and the
term is
. Find the A.P.
From the question it I s given that, \[{{T}_{10}}~=\text{ }41\] \[{{T}_{10}}~=\text{ }a\text{ }+\text{ }9d\text{ }=\text{ }41\]… [equation (i)] \[\begin{array}{*{35}{l}} {{T}_{15}}~=\text{ }a\text{...
If the seventh term of an A.P. is
and its ninth term is
, find its
term.
From the question it is given that, \[\begin{array}{*{35}{l}} {{T}_{9}}~=\text{ }1/7 \\ {{T}_{7}}~=\text{ }1/9 \\ \end{array}\] Let us assume ‘a’ be the first term and ‘d’ be the common...
Find the
term of an A.P. whose
term is
and
term is
.
From the question it is given that, \[\begin{array}{*{35}{l}} {{T}_{11}}~=\text{ }38 \\ {{T}_{6}}~=\text{ }73 \\ \end{array}\] Let us assume ‘a’ be the first term and ‘d’ be the common difference,...
Find the
term of the A.P. whose
term is
less than the
term, first term being
.
From the question it is given that, First term a = \[12\] \[{{\mathbf{7}}^{\mathbf{th}}}\] term is 24 less than the \[{{11}^{th}}\] term = \[{{T}_{11}}~\text{ }{{T}_{7}}~=\text{ }24\]...
Determine the A.P. whose third term is
and the
term exceeds the
term by
.
From the question it is given that, \[{{T}_{3}}~=\text{ }16\] The 7th term exceeds the 5th term by \[12\text{ }=\text{ }{{T}_{7}}~\text{ }{{T}_{5}}~=\text{ }12\] We know that, \[{{T}_{n}}~=\text{...
Which term of the A.P.
is the first negative term ?
From the question, The first term a = \[53\] Then, difference d \[\begin{array}{*{35}{l}} =\text{ }48\text{ }\text{ }53\text{ }=\text{ }-5 \\ =\text{ }43\text{ }\text{ }48\text{ }=\text{ }-5 \\...
Find the sum of the two middle most terms of the A.P.
From the question, Last term (nth) = \[4\frac{1}{3}\] = \[13/3\] First term a = \[-4/3\] Then, difference \[\begin{array}{*{35}{l}} d\text{ }=\text{ }-1\text{ }\text{ }\left( -4/3 \right)\text{...
(i) Find the
th term from the last term of the A.P.
. (ii) Find the
from the end of the A.P.
.
Let us assume \[253\text{ }as\text{ }{{n}^{th}}~\] term. From the question, The first term a = \[3\] Then, difference d \[\begin{array}{*{35}{l}} =\text{ }8\text{ }\text{ }3\text{ }=\text{ }5 \\...
(i) Check whether
is a term of the A.P.
(ii) Find whether 55 is a term of the A.P.
or not. If yes, find which term is it.
From the question it is given that, The first term a = \[11\] Then, difference d = \[8\text{ }\text{ }11\text{ }=\text{ }-3\] \[\begin{array}{*{35}{l}} 5\text{ }\text{ }8\text{ }=\text{ }-3 \\...
Which term of the A.P. (i)
? (ii)
?
Let us assume \[78\text{ }as\text{ }{{n}^{th}}~\] term. From the question, The first term a = \[3\] Then, difference d \[\begin{array}{*{35}{l}} ~=\text{ }8\text{ }\text{ }3\text{ }=\text{ }5 \\...
(i)If the common difference of an A.P. is
and the
term is
, then find its first term. (ii) If the first term of an A.P. is
and its
th term is zero, then find its common difference.
From the question it is given that, The \[\mathbf{1}{{\mathbf{8}}^{\mathbf{th}}}\] term = \[-5\] Then, common difference d = \[-3\] \[\begin{array}{*{35}{l}} {{T}_{n}}~=\text{ }a\text{ }+\text{...
Find the
term and the
term of the list of numbers:
From the question, The first term a = \[5\] Then, difference d = \[2\text{ }\text{ }5\text{ }=\text{ }\text{ }3\] \[\begin{array}{*{35}{l}} -1\text{ }\text{ }3\text{ }=\text{ }-3 \\ \text{ }4\text{...
Find the indicated terms in each of following A.P.s: (i)
(ii)
From the question, The first term a = \[1\] Then, difference d = \[6\text{ }\text{ }1\text{ }=\text{ }5\] \[\begin{array}{*{35}{l}} 11\text{ }\text{ }6\text{ }=\text{ }5 \\ 16\text{ }\text{...
Find the A.P. whose nth term is
. Also find the
th term.
From the question it is given that, nth term is \[\mathbf{7}\text{ }\text{ }\mathbf{3K}\] So, \[{{T}_{n}}~=\text{ }7\text{ }\text{ }3n\] Now, we start giving values, \[1,\text{ }2,\text{ }3,\text{...
Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms: (v)
(vi)
From the question it is given that, First term a = \[-10\] Then, difference d = \[-6\text{ }\text{ }\left( -\text{ }10 \right)\text{ }=\text{ }\text{ }6\text{ }+\text{ }10\text{ }=\text{ }4\]...
Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms: (iii)
(iv)
From the question it is given that, First term a = \[2\] Then, difference d = \[4\text{ }\text{ }2\text{ }=\text{ }2\] \[\begin{array}{*{35}{l}} 8\text{ }\text{ }4\text{ }=\text{ }4 \\ 16\text{...
Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms: (i)
(ii)
From the question it is given that, First term a = \[4\] Then, difference d = \[10\text{ }\text{ }4\text{ }=\text{ }6\] \[\begin{array}{*{35}{l}} 16\text{ }\text{ }10\text{ }=\text{ }6 \\ 22\text{...
Write first four terms of the A.P., when the first term a and the common difference d are given as follows: (iii)
(iv)
From the question it is given that, First term a = \[4\] Common difference d = \[-3\] Then the first four terms are = \[4\text{ }+\text{ }\left( -3 \right)\text{ }=\text{ }4\text{ }\text{ }3\text{...
Write first four terms of the A.P., when the first term a and the common difference d are given as follows: (i)
(ii)
From the question it is given that, First term a = \[10\] Common difference d = \[10\] Then the first four terms are = \[10+10=20\] \[\begin{array}{*{35}{l}} 20\text{ }+\text{ }10\text{ }=\text{...
For the following A.P.s, write the first term ‘a’ and the common difference ‘d’: (iii)
From the question, The first term a = \[-3.2\] Then, difference d = -3 – (-3.2) = -3 + 3.2 = 0.2 \[\begin{array}{*{35}{l}} -2.8\text{ }\text{ }\left( -3 \right)\text{ }=\text{ }-2.8\text{ }+\text{...
For the following A.P.s, write the first term ‘a’ and the common difference ‘d’: (i)
(ii)
From the question, The first term a = \[3\] Then, difference d = \[1\text{ }\text{ }3\text{ }=\text{ }\text{ }2\] \[\begin{array}{*{35}{l}} \text{ }1\text{ }\text{ }1\text{ }=\text{ }\text{ }2 \\...