Question

Find the sum of the GP :
(i) `sqrt(7)+sqrt(21)+3sqrt(7)+...` to n terms
(ii) `1-1/3+1/3^(2)-1/3^(3)+...` to n terms
(iii) `1-a+a^(2)-a^(3)+...` to n terms `(a ne 1)`
(iv) `x^(3)+x^(5)+x^(7)+...` to terms
(v) `x(x+y)+x^(2)(x^(2)+y^(2))+x^(3) (x^(3)+y^(3))+...` to n terms

Answer 1

Correct Answer - (i) `sqrt(7)/2 (sqrt(3)+1)(3^(n//2)-1)` (ii) `({3^(n)-(-1)^(n)})/(4xx3^((n-1)))` (iii) `({1-(-a)^(n)})/((1+a))`
(iv) `(x^(3)xx(x^(2n)-1))/((x^(2)-1))` (v) `(x^(2)(x^(2n)-1))/((x^(2)-1))+(xy(x^(n)y^(n)-1))/((xy-1))`
(i) `a=sqrt(7) and r=sqrt(21)/sqrt(7)=sqrt(3)`
`:. S_(n)=(sqrt(7)xx{(sqrt(3))^(n)-1})/((sqrt(3)-1))xx((sqrt(3)+1))/((sqrt(3)+1))=sqrt(7)/2.(sqrt(3)+1)(3^(n//2)-1)`
(ii) `a=1` and `r=(-1)/3`.
`:. S_(n)=(1xx{1-((-1)/3)^(n)})/((1+1/3))=3/4xx{1-((-1)^(n))/3^(n)}=({3^(n)-(-1)^(n)})/(4xx3^((n-1)))`.
(iii) `A=1 and R=-a`.
`:. S_(n)=(Axx{1-R^(n)})/((1-R))=(1xx{1-(-a)^(n)})/((1+a))=({1-(-a)^(n)})/((1+a))`
(iv) `a=x^(3) and r=x^(5)/x^(3)=x^(2)`.
`:. S_(n)=(x^(3)xx{(x^(2))^(n)-1})/((x^(2)-1))=(x^(3)xx(x^(2n)-1))/((x^(2)-1))`
(v) Given expression
`={x^(2)+x^(4)+x^(6)+..."to n terms"}+{xy+x^(2)y^(2)+x^(3)y^(3)+..."to n terms"}`
`=x^(2) {1+x^(2)+x^(4)+..."to n terms"}+xy{1+xy+x^(2)y^(2)+..."to n terms"}`
`=(x^(2)xx1xx{(x^(2))^(n)-1})/((x^(2)-1))+(xyxx1xx{(xy)^(n)-1})/((xy-1))=(x^(2)(x^(2n)-1))/((x^(2)-1))+(xy(x^(n)y^(n)-1))/((xy-1))`.