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Find the value of S= 2+6 4¹⁰⁰ + 2+2(6) 4⁹⁹ + 2+3(6) 4⁹⁸ + + 2+99(6) 4^2 + 2+100(6) 4 .

Correct answer

200

Step-by-step solution

aligned S & = 2+6 4¹⁰⁰ + 2+2(6) 4⁹⁹ + 2+3(6) 4⁹⁸ + + 2+99(6) 4^2 + 2+100(6) 4 & = 1 4¹⁰⁰ ((2+6)+(2+2(6)) 4+(2+3(6)) 4^2+ +(2+100(6)) 4⁹⁹ ) & = 1 4¹⁰⁰ (2 _ n=0 ⁹⁹ 4^n+ _ n=1 ¹⁰⁰ n 4^ n-1 ) G.P. and A.G.P. & = 1 4¹⁰⁰ (2 4¹⁰⁰-1 4-1 +6 ( 1-100 (4¹⁰⁰ ) 1-4 + 4 (1-4⁹⁹ ) (1-4)^2 ) )=200 aligned

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