Prove that $\displaystyle (1+2+2^2+2^3+...+2^200) \vdots 7$
note :there is +...+2^200 ,problem with latex
Hello, beq!x !
What does the \vdots mean? . . . "is divisible by" ?
If so, I have a proof . . .
Prove that: .$\displaystyle 1+2+2^2+2^3+ \hdots +2^{200}$ .is divisible by 7.
$\displaystyle N \;=\;(1 + 2 + 2^2) + (2^3+2^4+2^5) + (2^6+2^7+2^8) + \cdots + (2^{198} + 2^{199} + 2^{200}) $
. . $\displaystyle =\;(1+2+4) + 2^3(1+2+4) + 2^6(1+2+4) + \cdots + 2^{198}(1+2+4) $
. . $\displaystyle =\; \underbrace{7\left(1 + 2^3 + 2^6 + 2^9 + \cdots + 2^{198}\right)}_{\text{a multiple of 7}} $
Therefore, $\displaystyle N$ is divisible by 7.