Generating function doubt

Question

Please give me clarification

Answer 1

$\frac{1}{2}$$\sum_{r=0}^{20}(-1)^r(r+2)(r+1)=\frac{1}{2}\{(-1)^0(2)(1)+(-1)^1(3)(2)+(-1)^2(3)(4)+........+(-1)^{19}(21)(20)+(-1)^{20}(22)(21)\}$

                                                     $=\frac{1}{2}\{2*1-3*2+4*3-5*4+......-21*20+22*21\} $

                                                     $=\frac{1}{2}\{2(1-3)+4(3-5)+.......+20(19-21)+22*21\}$

                                                     $=\frac{1}{2}\{2(-2)+4(-2)+.....+20(-2)+22*21\}$

                                                     $=\frac{1}{2}\{(-2)(2+4+6+.....20)+22*21\}$

                                                      $=\frac{1}{2}\{(-2)(2(1+2+3+....+10))+22*21\}$

                                                      $=\frac{1}{2}\{(-4)(\frac{10*11}{2})+22*21\}$

                                                      $=\frac{1}{2}((-2)(110)+462)$

                                                       $=\frac{1}{2}(-220+462)$

                                                       $=\frac{1}{2}*242$

                                                        $=121$

Comments

@Sukannya , You forgot to divide by 2...Original question contains  1/2 outside..

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Yupp, thanks @ankItgupta.1729

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@sumit goyal 1 and Sukannya 

please see this

https://gateoverflow.in/205400/self-doubt-generating-function