InterviewSolution
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InterviewSolution
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`1.3+2.3^2+3.3^3+..............+n.3^n=((2n-1)3^(n+1)+3)/4`A. n, 2B. n, 3C. n + 1, 2D. n + 1, 3 |
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Answer» Correct Answer - D `1.3 + 2.3^(2)+3.3^(3)+...+n.3^(n)=((2n-1)3^(a)+b)/(4)` Let us put 3 = x. L.H.S: S = `x + 2x^(2)+3x^(3)+...+n.x^(n)" "...(1)` `xs = x^(2)+2x^(3)+3x^(4)+...+n.x^(n+1)" "...(2)` `(1)-(2) rArr S - xS = (x + 2x^(2)+3x^(2)+...+n.x^(n))-(x^(2)+2x^(3)+3x^(4)+...+n.x^(n+1))` `rArr S(1-x)=x+x^(2)+x^(3)+...+x^(n)-nx^(n+1)` `S(1-x)=(x(1-x^(n)))/(1-x)-nx^(n+1)` `rArr S=((1)/(x-1))((-x(x^(n)-1)+nx^(n+1)(x-1))/(x-1))` Put x = 3, `rArr S = (1)/(2)((-3^(n+1)+3+2n.3^(n+1))/(2))=((3^(n+1)(2n-1)+3)/(4))` |
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