If 1 + x + x 2 20 = ∑ r = 0 40 a r · x r , then ∑ r = 0 39 - 1 r · a r · a r + 1 equal to

Step-by-step solution

We have, 1 + x + x 2 20 = ∑ r = 0 40 a r x r x is replaced by - 1 x , we get 1 - 1 x + 1 x 2 20 = ∑ r = 0 40 ( - 1 ) r a r 1 x r 1 - x + x 2 20 = ∑ r = 0 40 ( - 1 ) r a r x 40 - r ∑ r = 0 39 - 1 r a r · a r + 1 is coefficient of x in product of 1 + x + x 2 20 1 - 1 x + 1 x 2 20 Now, 1 + x + x 2 20 1 - 1 x + 1 x 2 20 = 1 + x 2 + 1 x 2 20 Here, there will be no term of x . Hence, the coefficient of x will be 0 .