Question

find the value of the sum \\(\sum\limits_{i=5}^{7} \frac{8}{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:
$\frac{8}{5-1} + \frac{8}{6-1} + \frac{8}{7-1}$

Step2: Simplify each fraction

Calculate denominators and reduce fractions:
$\frac{8}{4} + \frac{8}{5} + \frac{8}{6} = 2 + \frac{8}{5} + \frac{4}{3}$

Step3: Find common denominator

Use 15 as common denominator to add:
$2 = \frac{30}{15}$, $\frac{8}{5} = \frac{24}{15}$, $\frac{4}{3} = \frac{20}{15}$

Step4: Sum the fractions

Add the numerators over common denominator:
$\frac{30 + 24 + 20}{15} = \frac{74}{15}$

Answer:

$\frac{74}{15}$