Question

Determinati ultima cifra a numarului n, unde : n = 1+1+2+2^2+2^3+...+2^2018

Answer 1

Notezi S=1+2+2^2+....+2^2018

2S=2+2^2+....+2^2018+2^2019

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2S-S=2+2^2+,,,,+2^2018+2^2019-1-2-,,,-2^2018=2^2019-1

n=1+2^2019-1=2^2019==2^(2016+3)

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Determini   ultima    cifra    a    lui  2^k

2^1=2

2^2=4

2^3=8

2^4=16 u(16)=6

2^5=32    U(32)=2

2^6=64    U(64)=4

2^5=128   U(128=8

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Observi    ca      ultima   cifra   se    repeta     din     4     in     4

Un   numar   de    forma   2^(4k+3)   are    ultima   cifra    8   unde   4k=2016

Deci  U(n)=8