Question

2 la puterea 1+2 la puterea 2+2 la puterea 3+...+2 la puterea 2014. Va rog ajutati-ma!

Answer 1

S = 2¹ + 2² + 2³ + ... + 2²⁰¹⁴  (·2)

2·S = 2² + 2³ + 2⁴ + ... + 2²⁰¹⁵

2·S - S = 2² + 2³ + 2⁴ + ... + 2²⁰¹⁵ - (2¹ + 2² + 2³ + ... + 2²⁰¹⁴)

S = -(2¹) + (2²-2²) + (2³-2³) + ... + (2²⁰¹⁴-2²⁰¹⁴) + 2²⁰¹⁵

S = 2²⁰¹⁵ - 2¹

S = 2²⁰¹⁵ - 2

Answer 2

$S = 2^1+2^2+2^3+...+2^{2013}+2^{2014}\big|\cdot 2 \\ \\ 2\cdot S = 2^2+2^3+2^4+...+2^{2014}+2^{2015} \\ \\ 2\cdot S = 2^1+2^2+2^3+2^4+...+2^{2014} +2^{2015} -2^1 \\ \\ 2\cdot S = S+2^{2015}-2^1 \\ \\ 2\cdot S-S = 2^{2015}-2 \\ \\ \Rightarrow S = 2^{2015} - 2$