Let C r denote the binomial coefficient of x r in the expansion of 1 + x 10 . If for α , β ∈ R , C 1 + 3 · 2 C 2 + 5 · 3 C 3 + … upto 10 terms = α × 2 11 2 β - 1 ( C 0 + C 1 2 + C 2 3 + … upto 10 terms) then the value of α + β is equal to _____.

Step-by-step solution

1 + x 10 = C 0 + C 1 x + C 2 x 2 + … … + C 10 x 10 Differentiating, we get 10 1 + x 9 = C 1 + 2 C 2 x + 3 C 3 x 2 + … + 10 C 10 x 9 Now replacing x → x 2 10 1 + x 2 9 = C 1 + 2 C 2 x 2 + 3 C 3 x 4 + … . + 10 C 10 x 18 Multiplying by x on both the sides 10 · x 1 + x 2 9 = C 1 x + 2 C 2 x 3 + 3 C 3 x 5 + … + 10 C 10 x 19 Differentiating again, we get 10 1 + x 2 9 · 1 + 9 1 + x 2 8 2 x = C 1 + 2 C 2 · 3 x 2 + 3 · 5 · C 3 x 4 + … + 10 · 19 C 10 x 1