Question

Determinati ultima cifra a numarului n , unde : n = 1 + 1 + 2 + 2²  + 2³ + . . . + 2²°°⁴ .

Answer 1

n = 1 + 1 + 2 + 2² + 2³ + . . . + 2²⁰⁰⁴ = 
= 1 +  (1 + 2 + 2² + 2³ + . . . + 2²⁰⁰⁴) =
= 1 +  (2⁰ + 2¹ + 2² + 2³ + . . . + 2²⁰⁰⁴) =
= 1 + (2²⁰⁰⁵ - 1) =
= 1 + 2²⁰⁰⁵ - 1 = 
= 2²⁰⁰⁵

U = ultima cifra

U(2²⁰⁰⁵) = U(2²⁰⁰⁴ ⁺ ¹) = U(2²⁰⁰⁴ × 2¹) = U(2⁴ * ⁵⁰¹ × 2) =
= U[(2⁴)⁵⁰¹ × 2] = U(16⁵⁰¹ × 2) = U(6 × 2 ) =  U(12) = 2