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Home » 14. The  term of an A.P. is three times the first and the  term exceeds twice the third term by Find the first term and the common difference.
14. The {{4}^{th}} term of an A.P. is three times the first and the {{7}^{th}} term exceeds twice the third term by 1. Find the first term and the common difference.
Arithmetic Progression | Class 10 | ICSE | Maths | RD Sharma
14. The {{4}^{th}} term of an A.P. is three times the first and the {{7}^{th}} term exceeds twice the third term by 1. Find the first term and the common difference.

An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant.

Solution:

Let’s consider the first term and the common difference of the A.P to be a and d respectively.

Then, we know that {{a}_{n}}=a+\left( n-1 \right)d

Given conditions,

{{4}^{th}} term of an A.P. is three times the first

Expressing this by equation we have,

⇒ {{a}_{4}}=3\left( a \right)

a+\left( 4-1 \right)d=3a

3d=2a ⇒ a=3d/2…….\left( i \right)

And,

{{7}^{th}} term exceeds twice the third term by 1

⇒ {{a}_{7}}=2\left( {{a}_{3}} \right)+1

a+\left( 7-1 \right)d=2\left( a+\left( 3-1 \right)d \right)+1

a-2d+1=0 ….. \left( ii \right)

Using \left( i \right) in \left( ii \right), we have

3d/2-2d+1=0

3d-4d+2=0

d=2

So, putting d=2 in \left( i \right), we get a

⇒ a=3

Therefore, the first term is 3 and the common difference is 2.

i

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