Problem 41 - College Algebra

Question

Find each indicated sum. $$\sum_{i=1}^{5} \frac{i !}{(i-1) !}$$

Step-by-Step Solution

Answer

The result of the sum \(\sum_{i=1}^{5} \frac{i !}{(i-1) !}\) is 15.

Step 1: Parsing the series

The provided series is \(\sum_{i=1}^{5} \frac{i !}{(i-1) !}\). In this summation, \(i\) takes on values from 1 to 5 and for each \(i\), the term \(\frac{i !}{(i-1) !}\) is calculated and added to the sum.

Step 2: Understanding Factorial Division

They key to solving this exercise is to understand that by definition \(i ! = i * (i-1) *\dots*2*1\), so \(\frac{i !}{(i-1) !} = i\).

Step 3: Substituting and Calculating

After substituting the expression obtained in the previous step the series becomes \(\sum_{i=1}^{5} i\), resulting in the result being \(1 + 2 + 3 + 4 + 5 = 15\).