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\begin{document}
\title{What is $7^1+7^2+7^3+7^4+7^5$?}
\author{Amit Yaron}
\date{Sep 16, 2025}
\maketitle
Another one from YouTube that you are not supposed to solve it the naive way.
The way to solve it is similar to that you use for solving a quartic equation
wit symmetry, such as:
\[
  x^4+x^3+x^2+x+1=0
\]
So, I'm gonna divide the expression by $7^3$ and multiply by it. Notice that
$7+\frac17=\frac{50}7$, and here 50 is to be raised to the second power.
Let's start:
\begin{align*}
  7^1+7^2+7^3+7^4+7^5&=7^3\bigg(\frac1{7^2}+\frac17+1+7+7^2\big)\\
  &=7^3\Bigg(\bigg(7+\frac17\bigg))+\bigg(7^2+\frac1{7^2}\bigg)+1\Bigg)\\
  &=7^3\Bigg(\bigg(7+\frac17\bigg))\\&\quad+\bigg(7^2+\frac1{7^2}i+2\cdot7\cdot\frac17-2\cdot7\cdot\frac17\bigg)+1\Bigg)\\
  &=7^3\Bigg(\bigg(7+\frac17\bigg)+{\bigg(7+\frac17\bigg)}^2-2+1\Bigg)\\
  &=7^3\bigg(\frac{50}7+\frac{50^2}{7^2}-1\bigg)\\
  &=7^3\bigg(\frac{7\cdot50+50^2-49}{49}\bigg)\\
  &=7(350+2500-49)\\
  &=7(2500+301)\\
  &=17500+2107\\
  &=19607
\end{align*}
Cool! Without even cubing the digit 7 or anything else. Easier than using a
calculator!
\end{document}
