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Find the sum of the following arithmetic progressions:(v) a+b, a-b, a-3b, … to 22 terms (vi) {{\left( x-y \right)}^{2}},\left( {{x}^{2}}+{{y}^{2}} \right),{{\left( x+y \right)}^{2}} to 22 terms
Arithmetic Progressions | CBSE | Class 10 | Exercise 9.6 | Maths | RD Sharma
Find the sum of the following arithmetic progressions:(v) a+b, a-b, a-3b, … to 22 terms (vi) {{\left( x-y \right)}^{2}},\left( {{x}^{2}}+{{y}^{2}} \right),{{\left( x+y \right)}^{2}} to 22 terms

(v) a+b, a-b, a-3b, ….. to 22 terms

First term (a) =a+b

Common difference (d) ={{a}_{n}}-{{a}_{n-1}}=-b-a-b=-2b

Sum of n terms of A.P is {{S}_{n}}=n/2\left\{ 2a\left( n-1 \right)d \right\}, where n=22

{{S}_{22}}=22/2\left[ 2\left( a+b \right)+\left( 22-1 \right)-2b \right]

=11{2(a+b)-22b)

=11{2a-20b}

=22a-440b

\therefore {{S}_{22}}=22a-440b

(vi) {{\left( x-y \right)}^{2}},\left( {{x}^{2}}+{{y}^{2}} \right),{{\left( x+y \right)}^{2}}….. to n terms

First term (a) ={{\left( x-y \right)}^{2}}

Common difference (d) ={{a}_{n}}-{{a}_{n-1}}={{x}^{2}}+{{y}^{2}}-{{\left( x-y \right)}^{2}}

={{x}^{2}}+{{y}^{2}}-\left( {{x}^{2}}+{{y}^{2}}-2xy \right)

={{x}^{2}}+{{y}^{2}}-{{x}^{2}}+{{y}^{2}}+2xy

=2xy

Sum of n terms of A.P is {{S}_{n}}=n/2\left[ 2a+\left( n-1 \right)d \right]

=n/2\left[ 2{{\left( x-y \right)}^{2}}+\left( n-1 \right)2xy \right]

=n\left\{ {{\left( x-y \right)}^{2}}+\left( n-1 \right)xy \right\}

\therefore {{S}_{n}}=n\left\{ {{\left( x-y \right)}^{2}}+\left( n-1 \right)xy \right\}

i

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