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Question
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
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Solution
(x + y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ...to n terms;
Let Sn = (x + y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ...to n terms
Let us multiply and divide by (x – y) we get,
Sn = `1/(x – y)` [(x + y)(x – y) + (x2 + xy + y2)(x – y) ...upto n terms]
(x – y)Sn = (x2 – y2) + x3 + x2y + xy2 – x2y – xy2 – y3 ...upto n terms
(x – y)Sn = (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))}`
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