Geometric Sequences

Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This common ratio is denoted by the letter r.

Definition of a Geometric Sequence

A geometric sequence can be expressed as:

a, ar, ar^2, ar^3, ..., ar^(n-1)

where:

• a = the first term of the sequence
• r = the common ratio
• n = the number of terms in the sequence

Finding the nth Term

The formula to find the nth term (Tn) of a geometric sequence is:

Tn = ar^(n-1)

where Tn is the nth term, a is the first term, r is the common ratio, and n is the term number.

Example 1: Finding the nth Term

Consider a geometric sequence where the first term (a) is 3 and the common ratio (r) is 2. To find the 5th term (T5), use the formula:

T5 = 3 × 2^(5-1)

Calculate:

T5 = 3 × 2^4

T5 = 3 × 16

T5 = 48

The 5th term of the sequence is 48.

Sum of the First n Terms

The sum of the first n terms (S_n) of a geometric sequence can be calculated using the formula:

S_n = a(1 - r^n) / (1 - r) for r ≠ 1

where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.

Example 2: Finding the Sum of the First n Terms

Using the same geometric sequence where a = 3 and r = 2, find the sum of the first 5 terms (S_5).

Using the formula:

S_5 = 3(1 - 2^5) / (1 - 2)

Calculate:

S_5 = 3(1 - 32) / (-1)

S_5 = 3(-31) / (-1)

S_5 = 93

The sum of the first 5 terms of the sequence is 93.

Infinite Geometric Series

An infinite geometric series is the sum of the terms of a geometric sequence that continues indefinitely. This series converges only if the absolute value of the common ratio is less than 1 (|r| < 1). The formula for the sum (S) of an infinite geometric series is:

S = a / (1 - r)

Example 3: Sum of an Infinite Geometric Series

Consider an infinite geometric series where the first term (a) is 5 and the common ratio (r) is 0.5. To find the sum:

S = 5 / (1 - 0.5)

Calculate:

S = 5 / 0.5

S = 10

The sum of the infinite geometric series is 10.

Applications of Geometric Sequences

Geometric sequences have various applications in real life. They are used in finance to calculate compound interest, in computer science for algorithm analysis, and in physics for exponential decay processes.

Example 4: Compound Interest

Suppose you invest R1,000 at an interest rate of 5% per year compounded annually. The amount of money after n years can be represented as a geometric sequence where:

• a = R1,000
• r = 1 + 0.05 = 1.05

The amount after 3 years (A3) can be calculated as:

A3 = 1000 × 1.05^(3)

Calculate:

A3 = 1000 × 1.157625

A3 ≈ 1157.63

After 3 years, the amount will be approximately R1,157.63.

Summary

• A geometric sequence is defined by a first term and a common ratio.
• The nth term can be found using Tn = ar^(n-1).
• The sum of the first n terms is S_n = a(1 - r^n) / (1 - r).
• An infinite geometric series converges if |r| < 1, with the sum S = a / (1 - r).

Check your understanding

1. What is the common ratio in the geometric sequence 4, 12, 36, 108?
2. Calculate the 6th term of the geometric sequence where a = 5 and r = 3.
3. Find the sum of the first 4 terms of the geometric sequence with a = 2 and r = 0.5.
4. Determine if the infinite geometric series with a = 10 and r = 0.2 converges, and if so, find its sum.