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Question

The value of $sum from k equals 1 to 100 of sin space left parenthesis k x right parenthesis cos space left parenthesis 101 minus k right parenthesis x$ is equal to

$\frac{101}{2} \sin (101 x)$
$99 \sin (101 x)$
$50 \sin (101 x)$
$100 \sin (101 x)$

hint Hint:

In this question, we have to find the value of $sum from k equals 1 to 100 of sin space left parenthesis k x right parenthesis cos space left parenthesis 101 minus k right parenthesis x$ . For that we will write the sum in both actual and the reverse order. Further, we will add both the equation and then solve the equation obtained to find the sum.

The correct answer is: $50 sin space left parenthesis 101 x right parenthesis$

$sum from k equals 1 to 100 of sin space left parenthesis k x right parenthesis cos space left parenthesis 101 minus k right parenthesis x S equals sin space x. cos space 100 x space plus space sin space 2 x. cos space 99 x space plus space sin space 3 x. cos space 98 x space plus..... plus sin space 100 x. cos space x space _ _ _ _ _ _ _ _ e q u a t i o n space 2 W r i t t i n g space i n space t h e space r e v e r s e space o r d e r comma space S equals sin space 100 x. cos space x plus space sin space 99 x. cos space 2 x space plus space sin space 98 x. cos space 3 x space plus...... plus sin space x. cos space 100 x space space space _ _ _ _ _ _ _ _ e q u a t i o n space 1 A d d i n g space t h e space e q u a t i o n space 1 space a n d space 2 comma space w e space g e t colon rightwards double arrow 2 S equals left square bracket left parenthesis sin space x. cos space 100 x space plus space sin space 100 x. cos space x right parenthesis plus left parenthesis sin space 2 x. cos space 99 x space plus space sin space 99 x. cos space 2 x right parenthesis plus............ plus left parenthesis sin space 1 o o x. cos space x space plus space sin space x. cos space 100 x right parenthesis right square bracket rightwards double arrow 2 S equals 100 sin left parenthesis x plus 100 x right parenthesis rightwards double arrow S equals 50 sin left parenthesis 101 x right parenthesis$

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