Question

find the value of the sum \\(\sum\limits_{i = 5}^{7}\frac{20}{i - 1}\\). use a summation formula, if possible. write the exact answer. do not round.

Explanation:

Step1: Expand the summation

Substitute $i=5,6,7$ into the term:
$\sum_{i=5}^{7} \frac{20}{i-1} = \frac{20}{5-1} + \frac{20}{6-1} + \frac{20}{7-1}$

Step2: Simplify each fraction

Calculate the denominator for each term:
$= \frac{20}{4} + \frac{20}{5} + \frac{20}{6}$

Step3: Simplify individual terms

Reduce each fraction to simplest form:
$= 5 + 4 + \frac{10}{3}$

Step4: Sum the terms

Convert to common denominator and add:
$= 9 + \frac{10}{3} = \frac{27}{3} + \frac{10}{3} = \frac{37}{3}$

Answer:

$\frac{37}{3}$