InterviewSolution
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InterviewSolution
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Statement-1: `(C_(0))/(2.3)- (C_(1))/(3.4) +(C_(2))/(4.5)-.............+............+(-1)^(n) (C_(n))/((n+2)(n+3))= (1)/((n+1)(n+2))` Statement-2:`(C_(0))/(k)- (C_(1))/(k+1) +(C_(2))/(k+3)+............+(-1)^(n) (C_(n))/(k+n)=int_(0)^(1)x^(k-1) (1 - x)^(n) dx`A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - a We have, `(1 - x)^(n) = C_(0) - C_(1) x + C_(2) x^(2) - ...+ (-1)^(n) C_(n)x^(4)` `rArr x^(k-1) (1 - x)^(n) = C_(0) x^(k-1) - C_(1) x^(k)+ C_(2) x^(k+1)-.... + (-1)""^(n)C_(n) x^(n+k-1) ` `rArr underset(0)overset(1)(int)x^(k-1) (1 - x)^(n) dx=(C_(0))/(k)-(C_(1))/(k+1) + (C_(2))/(k+2) - ....+ (-1)^(n) (C_(n))/(k+n)` So, statement-2 is true. Replacing k= 2 and k = 3 respecitvely, we get `underset(0)overset(1)(int)x^(k-1) (1 - x)^(n) dx=(C_(0))/(2)-(C_(1))/(3) + (C_(2))/(4) - ....+ (-1)^(n) (C_(n))/(n+2)` ...(i) and `underset(0)overset(1)(int)x (1 - x)^(n) dx=(C_(0))/(3)-(C_(1))/(4) + (C_(2))/(5) - ....+ (-1)^(n) (C_(n))/(n+2)` ...(ii) Subtracting (ii) from (i), we get `underset(0)overset(1)(int)x (1 - x)^(n+1) dx-(C_(0))/(2.3)-(C_(1))/(3.4) + (C_(2))/(4.5) - ....+ (-1)^(n) (C_(n))/((n+2)(n+3))` `rArr underset(0)overset(1)(int) (1 - x)x^(n+1) dx=(C_(0))/(2.3)-(C_(1))/(3.4) + (C_(2))/(4.5) - ...` `+ (-1)^(n) (C_(n))/((n+2)(n+3))[because underset(0)overset(a)(int) f(x) dx = underset(0)overset(a)(int)f(a-x)dx]` `underset(0)overset(1)(int) (x^(n+1)- x^(n+2)) dx=(C_(0))/(2.3)-(C_(1))/(3.4) + (C_(2))/(4.5)... + (-1)^(n) (C_(n))/((n+2)(n+3))` `rArr (1)/((n+1)(n+2))= (C_(0))/(2.3)-(C_(1))/(3.4) + (C_(2))/(4.5)-...+ (-1)^(n) (C_(n))/((n+2)(n+3))` Hence, statement-1 is also true and statement-2 is a correct expanation for statement-1 |
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