Question

Find the sum of the following geometric series:

(x +y) + (x² + xy + y²) + (x³ + x²y + xy² + y³) + ... to n terms;

Answer 1

(x + y) + (x² + xy + y²) + (x³ + x²y + xy² + y³) + ...to n terms;

Let S_n = (x + y) + (x² + xy + y²) + (x³ + x²y + xy² + y³) + ...to n terms

Let us multiply and divide by (x – y) we get,

S_n = `1/(x – y)` [(x + y)(x – y) + (x² + xy + y²)(x – y) ...upto n terms]

(x – y)S_n = (x² – y²) + x³ + x²y + xy² – x²y – xy² – y³ ...upto n terms

(x – y)S_n = (x² + x³ + x⁴ + ...n terms) – (y² + y³ + y⁴ +...n terms)

By using the formula,

Sum of GP for n terms = `(a(1 – r^n))/(1 – r)`

We have two G.Ps in the above sum, so,

`(x – y) S_n = x^2((x^n – 1)/(x – 1)) –  y^2((y^n – 1)/(y – 1))`

` S_n = 1/(x – y) . {x^2((x^n – 1)/(x – 1)) –  y^2((y^n – 1)/(y – 1))}`