Question

Arătaţi că A=3+3² +3³ + ... + 3²⁰⁰³ + 3²⁰¹⁶ este divizibil cu 4. urgeeeent!!​

Answer 1

Răspuns:

$\bf A=3+3^2 +3^2 + ... + 3^{2015} + 3^{2016}$

$A=3^{1}\cdot\big(3^{1-1}+3^{2-1} +3^{3-1} +3^{4-1}\big)+ ... + 3^{2013}\cdot\big(3^{2013-2013}+3^{2014-2013} +3^{2015-2013} +3^{2016-2013}\big)$

$A=3^{1}\cdot\big(3^{0}+3^{1} +3^{2} +3^{3}\big)+ ... + 3^{2013}\cdot\big(3^{0}+3^{1} +3^{2} +3^{3}\big)$

$A=3^{1}\cdot\big(1+3 +9 +27\big)+ ... + 3^{2013}\cdot\big(1+3 +9 +27\big)$

$A=3^{1}\cdot40+3^{5}\cdot40+3^{9}\cdot40+ .... + 3^{2013}\cdot40$

$A=40 \cdot\big(3^{1}+3^{5}+3^{9}+ .... + 3^{2013}\big)$

$\red{\boxed{A=4\cdot 10 \cdot\big(3^{1}+3^{5}+3^{9}+ .... + 3^{2013}\big)~~\vdots~~4~}}$