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in Geometric Progressions by (15.7k points)

Find the sum to n terms of the sequence :

(i) \(\big( {\text{x}} + \frac{1}{{\text{x}} }\big)^2\)\(\big( {\text{x}^2} + \frac{1}{{\text{x}^2} }\big)^2\)\(\big( {\text{x}^3} + \frac{1}{{\text{x}^3} }\big)^2\)

(ii) (x + y), 9x2 + xy + y2), (x3 + x2y + xy2 + y3), …. to n terms

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1 Answer

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by (15.2k points)

This can also be written as

\(\big( {\text{x}^2} + \frac{1}{{\text{x}^2}} + 2\big)\) + \(\big( {\text{x}^4} + \frac{1}{{\text{x}^4}} + 2\big)\) + \(\big( {\text{x}^6} + \frac{1}{{\text{x}^6}} + 2\big)\) + ....... to n term

Sum of a G.P. series is represented by the formula, Sn = a\(\frac{r^n - 1}{r-1}\), when r≠1. 

‘Sn’ represents the sum of the G.P. series up to nth terms, 

‘a’ represents the first term, 

‘r’ represents the common ratio and 

‘n’ represents the number of terms.

a = X2\(\frac{1}{{\text{x}} ^2}\)

r = (ratio between the n term and n-1 term)  X2\(\frac{1}{{\text{x}} ^2}\) 

n terms

(ii) If we divide and multiply the terms by (x-y)

Sum of a G.P. series is represented by the formula, Sn = \(a\frac{r^n - 1}{r-1}\), when r≠1. 

‘Sn’ represents the sum of the G.P. series up to nth terms, 

‘a’ represents the first term, 

‘r’ represents the common ratio and 

‘n’ represents the number of terms. 

Here,

a = x2 , y2 

r = (ratio between the n term and n-1 term) x, y

n terms

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