Question

1) Cmr : \(A=75\left(4^{2015}+4^{2014}+4^{2013}+....+4^2+5\right)+25\)chia hết cho \(4^{2016}\)

Answer 1

đặt B = 4²⁰¹⁵ + 4²⁰¹⁴ + 4²⁰¹³  + ... + 4²

4B = 4²⁰¹⁶ + 4²⁰¹⁵ + 4²⁰¹⁴ + ... + 4³

4B - B = ( 4²⁰¹⁶ + 4²⁰¹⁵ + 4²⁰¹⁴ + ... + 4³ ) - ( 4²⁰¹⁵ + 4²⁰¹⁴ + 4²⁰¹³  + ... + 4² )

3B = 4²⁰¹⁶ - 4²

\(\Rightarrow\)B = \(\frac{4^{2016}-4^2}{3}\)hay B = \(\frac{4^{2016}-16}{3}\)

\(\Rightarrow\)A = 75 . ( \(\frac{4^{2016}-16}{3}\)+ 5 ) + 25

A = 75 . ( \(\frac{4^{2016}-16}{3}\)+ \(\frac{15}{3}\)) + 25

A = 75 . ( \(\frac{4^{2016}-1}{3}\)) + 25

A = 25 . ( 3 . \(\frac{4^{2016}-1}{3}\)) + 25

A = 25 . ( 4²⁰¹⁶ - 1 ) + 25

A = 25 . ( 4²⁰¹⁶ - 1 + 1 )

A = 25 . 4²⁰¹⁶ \(⋮\)4²⁰¹⁶