Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 5 - Sequences, Mathematical Induction, and Recursion - Exercise Set 5.2: 27

Answer

$5\times\frac{5^{k}-25}{4} $

Work Step by Step

We know that, $1 + r + r^2 + ··· + r^n = \frac{r^{n+1}− 1}{r − 1}$ $5^3 + 5^4 + 5^5 + ··· + 5^k $ = $5^3(1+5^1 + 5^2 + ··· + 5^{k-3}) $ =$5^3\times\frac{5^{(k-3)+1}-1}{5-1} $ =$5^3\times\frac{5^{k-2}-1}{4} $ or $\frac{5^{k+1}-5^3}{4} $ =$5\times\frac{5^{k}-5^2}{4} $
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