(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 ${{S}_{n}}=n/2\left[ 2a+\left( n-1 \right)d \right]$

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

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

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