Unit 4: The Sum of Geometric Series

A geometric series is the sum of the terms of a geometric sequence. It can be either finite (with a limited number of terms) or infinite (adding terms endlessly).

Sum of the First \( n \) Terms (Finite Geometric Series)

For a geometric series where the first term is \( a \) and the common ratio is \( r \):

\( S_n = \dfrac{a(1 - r^n)}{1 - r},\quad r \ne 1 \)

Sum to Infinity (Infinite Geometric Series)

If \( -1 < r < 1 \), the infinite geometric series converges, and its sum is:

\( S_{\infty} = \dfrac{a}{1 - r} \)

1. Identify the first term (\( a \)) and the common ratio (\( r \)).
2. Determine the number of terms (\( n \)) for finite series or check if \( -1 < r < 1 \) for infinite series.
3. Use the appropriate formula to calculate the sum.

Example 1: Finite Series

Problem: Calculate the sum of the first 7 terms of the geometric series \( 3, 15, 75, \dots \).

Solution:

• First term, \( a = 3 \)
• Common ratio, \( r = \dfrac{15}{3} = 5 \)
• Number of terms, \( n = 7 \)

Using the formula:

\( S_n = \dfrac{a(1 - r^n)}{1 - r} \)

Substituting values:

\( S_7 = \dfrac{3(1 - 5^7)}{1 - 5} = \dfrac{3(1 - 78125)}{-4} \)

\( \dfrac{-234372}{-4} = 58,593 \)

Therefore, the sum of the first 7 terms is 58,593.

Example 2: Sum to Infinity

Problem: Find the sum to infinity of the series \( 16, 8, 4, \dots \).

Solution:

• First term, \( a = 16 \)
• Common ratio, \( r = \dfrac{8}{16} = 0.5 \)
• Since \( -1 < r < 1 \), the infinite sum exists.

Using the formula:

\( S_{\infty} = \dfrac{a}{1 - r} \)

Substituting values:

\( S_{\infty} = \dfrac{16}{1 - 0.5} = \dfrac{16}{0.5} = 32 \)

Therefore, the sum to infinity is 32.

1. Find the sum of the first 6 terms of the geometric series \( 2, 6, 18, \dots \).
2. If the first term is 5 and the common ratio is 3, find the sum to infinity if possible.
3. Determine the sum to infinity of the series \( 8, 4, 2, 1, \dots \).
4. The 4th term of a geometric series is 16, and the common ratio is 2. Find the sum of the first 5 terms.

1. What is the sum of the first 4 terms of the geometric series \( 1, 2, 4, 8, \dots \)?

15

16

30

31

2. What is the sum to infinity of the series \( 3, 1.5, 0.75, \dots \)?

5

6

7

8

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• A geometric series is the sum of the terms of a geometric sequence.
• The sum of a finite geometric series can be calculated using \( S_n = \dfrac{a(1 - r^n)}{1 - r} \).
• The sum to infinity exists when \( -1 < r < 1 \) and is calculated using \( S_{\infty} = \dfrac{a}{1 - r} \).