Let S = _ k=1 ⁵⁰ 2k-1 2^k . If S can be expressed in the form a - b 2⁵⁰ , where a and b are integers, find the value of a + b .

Step-by-step solution

Given S = _ k=1 ⁵⁰ 2k-1 2^k . Expanding the summation, we have: S = 1 2 + 3 2^2 + 5 2^3 + + 99 2⁵⁰ Multiplying both sides by 1 2 , we get: S 2 = 1 2^2 + 3 2^3 + 5 2^4 + + 97 2⁵⁰ + 99 2⁵¹ Subtracting the second equation from the first: S - S 2 = 1 2 + ( 2 2^2 + 2 2^3 + + 2 2⁵⁰ ) - 99 2⁵¹ S 2 = 1 2 + 2 ( 1 2^2 + 1 2^3 + + 1 2⁵⁰ ) - 99 2⁵¹ The terms in the parenthesis form a geometric progression with 49 terms, first term a = 1 4 , and common ratio r = 1 2 . Sum of this GP = 1 4 ( 1 - ( 1 2 )⁴⁹ ) 1 - 1 2 = 1 2 ( 1 - 1