Question

Calculati 1+6+6la2+...+6la2013=?

Answer 1

$S=6^0+6^1+6^2+...+6^{2013}$

Inmultim expresia cu 6:

$6S=6^1+6^2+...+6^{2013}+6^{2014}$

Dupa cum observi, toti termeni mai putin ultimul, sunt defapt S - 1:

[tex]6S=S-1+6^{2014}\\ 5S=6^{2014}-1\\\\ S= \frac{6^{2014}-1}{5} [/tex]

Answer 2

S= 1+n+n^2+...+n^k, inmultim cu n

nxS=nx(1+n+n^2+...+n^k)

nxS=n+n^2+n^3+...+n^(k+1), adunam 1

1+nxS=1+n+n^2+n^3+...+n^(k+1)

1+nxS=(1+n+n^2+n^3+...n^k)+n^(k+1)

1+nxS=S+n^k+1)

nxS-S=n^(k+1) -1

S(n-1)=n^(k+1) -1

S=[n^(k+1) -1]/(n-1)

In cazul nostru: n=6 si k=2013, rezulta S=(6^2014 -1)/5