Let S = _ k=1 ¹² k^2 2^k . If S is expressed in the form A 2¹³ - B , where A and B are positive integers, find the value of A + B .

Step-by-step solution

Given S = 1^2 2^1 + 2^2 2^2 + 3^2 2^3 + + 12^2 2¹² . Multiplying by 2 : 2S = 1^2 2^2 + 2^2 2^3 + + 11^2 2¹² + 12^2 2¹³ Subtracting 2S from S : -S = 1^2 2^1 + (2^2 - 1^2)2^2 + (3^2 - 2^2)2^3 + + (12^2 - 11^2)2¹² - 12^2 2¹³ -S = 2 + 3 2^2 + 5 2^3 + + 23 2¹² - 144 2¹³ Let S₁ = 3 2^2 + 5 2^3 + + 23 2¹² . This is an Arithmetico-Geometric Progression. Multiplying by 2 : 2S₁ = 3 2^3 + 5 2^4 + + 21 2¹² + 23 2¹³ Subtracting 2S₁ from S₁ : -S₁ = 3 2^2 + 2 2^3 + 2 2^4 + + 2 2¹² - 23 2¹³ -S₁ = 12 + 2(2^3 + 2^4 + + 2¹²) - 23 2¹³