Section 8.5: Applications & Review

This section provides applications of sequences and series in real-world problems, followed by review exercises for practice and assessment.

Example 1: Arithmetic Sequence Application

A savings account deposits $100 each month. How much money will be saved after 12 months?

Step 1: Use arithmetic series formula: $ S_n = \frac{n}{2}(a_1 + a_n) $

Step 2: Here: $ a_1 = 100, a_{12} = 100, n=12 $

Step 3: Compute: $ S_{12} = \frac{12}{2} (100+100) = 6 \cdot 200 = 1200 $

Result: $1,200 saved after 12 months

An investment grows 5% per year. If the initial amount is $1,000, find the total after 4 years.

Step 1: Use geometric sequence formula: $ a_n = a_1 \cdot r^{n-1} $

Step 2: Here: $ a_1 = 1000, r = 1.05, n = 4 $

Step 3: Compute: $ a_4 = 1000 \cdot (1.05)^3 \approx 1157.63 $

Result: Approx. $1,157.63 after 4 years

Find the sum of the first 20 terms of the arithmetic sequence: 3, 7, 11, ...
Find the 5th term of the geometric sequence: 2, 6, 18, ...
Find the sum of a geometric series: 3 + 6 + 12 + ... + 192
An auditorium has 10 rows with 5 more seats in each subsequent row. Total seats?
Determine the 6th term and the sum of the first 6 terms: 4, 7, 10, ...