InterviewSolution
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InterviewSolution
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`lim_(xrarr2)(sum_(r=1)^(n)x^r-sum_(r=1)^(n)2r)/(x-2)` is equal toA. nB. `(n-1)2^n`C. `(n-1)2^n+1`D. none of these |
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Answer» Correct Answer - C We have , `S=lim_(xto2)(sum_(r=1)^(n)x^r-sum_(r=1)^(n)2r)/(x-2)` `rArr S=lim_(xto2)(sum_(r=1)^(n)(x^r-2^r))/(x-2)` ` rArr S=sum_(r=1)^(n) lim_(xto2)((x^r-2^r)/(x-2))` ` rArr S=sum_(r=1)^(n)r.2^(r-1)` ` rArr S=1+2.2 +3.2^2+4.2^3+....+N.2^n-1` .....(i) ` rArr 2S=1.2+2.2^2+3.2^3+....+(n-1)2^(n-1)+n.2^n` .....(ii) Subtracting (i) from (ii), we get ` S=-1-1.2-1.2^2-1.2^3.....-1.2^n-1+n.2^n` `rArr S=-(1+2+2^2+.....+2^n-1)+n.2^n` `rArr S=-((2^n-1)/(2-1))+n.2^n=-2^n+1+n.2^n` |
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