_ n _ r=1 ^n ⁻¹ (r^2+ 3 4 )=

Step-by-step solution

aligned & _ n _ r=1 ^n ⁻¹ (r^2+ 3 4 ) & = _ n _ r=1 ^n ⁻¹ [ 1 r^2+1- 1 4 ] & = _ n _ r=1 ^n ⁻¹ [ (r+ 1 2 )- (r- 1 2 ) 1+ (r+ 1 2 ) (r- 1 2 ) ] & = _ n _ r=1 ^n [ ⁻¹ (r+ 1 2 )- ⁻¹ (r- 1 2 ) ], & [ ⁻¹(a-b)= ⁻¹ [ a-b 1+a b ] ] aligned aligned & = _ n [ ( ⁻¹ 3 2 - ⁻¹ ( 1 2 ) ) ]+ [ ⁻¹ 5 2 - ⁻¹ 3 2 ] & + [ ⁻¹ 7 2 - ⁻¹ 5 2 ] & + [ ⁻¹ (n+ 1 2 )- ⁻¹ (n- 1 2 ) ] & = _ n [ ⁻¹ (n+ 1 2 )- ⁻¹ ( 1 2 ) ] & = _ n [ ⁻¹ (n+ 1 2 )- ⁻¹ ( 1 2 ) ] & = ⁻¹ - ⁻¹ 1 2 = 2 - ⁻¹ 1 2 & = ⁻¹ 1 2 = ⁻¹ 2 aligned