Definition and Explanation
In a geometric sequence, each term is the product of the previous term and a constant called the common ratio, denoted by r. Here’s how it works:
1st term: a (starting value)
2nd term: a × r (multiply the first term by r)
3rd term: a × r × r (multiply the second term by r)
And so on…
Example: In the sequence 2, 6, 18, 54, the common ratio is 3, as each term is 3 times the previous term.
General Formula
The general formula for the n-th term of a geometric sequence is:
aₙ = a × r(n-1)
Explanation:
a = first term
r = common ratio (can be a fraction)
n = term number
(n-1) = number of times r is multiplied
Example with Fractional Common Ratio:
Consider the sequence 16, 8, 4, 2, where the common ratio is 1/2:
5th term = 16 × (1/2)⁴ = 16 × 1/2⁴ = 16 × 1/16 = 16/16 = 1
Working Through Examples
Let’s work through examples to understand the concept better.
Example 1: Finding the Common Ratio
Consider the sequence 5, 15, 45, 135:
r = 15 ÷ 5 = 3 (divide any term by the previous term)
Example 2: Using the General Formula
For the sequence 5, 10, 20, 40, find the 6th term:
a₆ =5 × 2⁽⁶⁻¹⁾ = 5 × 2⁵ = 5 × 32 = 160 (use the formula with a=5, r=2, n=6)
Application: Compound Interest
Geometric sequences are used to calculate compound interest in finance. Here’s how it works:
Initial Investment: P
Annual Interest Rate: r
Number of Years: n
The amount after n years, A, is given by the formula:
A = P × (1 + r)n
Example:
If you invest $1000 with an annual interest rate of 10%, compounded annually, for 3 years:
P = $1000, r = 0.10, n = 3
A = $1000 × (1 + 0.10)3 = $1000 × 1.103 = $1331
Your investment grows to $1331 in 3 years.
Derivation of the General Formula for a Geometric Sequence
Let’s explore why the formula for the n-th term of a geometric sequence is a × r(n-1) and not a × rn.
Consider a geometric sequence with a common ratio of r:
1st term: a (starting value)
2nd term: a × r (multiply the first term by r)
3rd term: a × r2 (multiply the second term by r)
4th term: a × r3 (multiply the third term by r)
And so on…
Notice that the exponent in each term is one less than the term number. This pattern continues for all terms in the sequence.
General Formula:
The n-th term can be expressed as:
an = a × r(n-1)
Explanation:
a = first term (starting value)
r = common ratio
n = term number
(n-1) = number of times r is multiplied (one less than the term number)
Example:
For the sequence 3, 6, 12, 24, the common ratio is 2, and the 4th term is:
a4 = 3 × 23 = 3 × 8 = 24
This derivation helps us understand why the formula includes r(n-1) and not rn, reflecting the pattern observed in geometric sequences.
Exponential Decay Sequence: (1/2)⁽ⁿ⁻¹⁾
This sequence is defined by the formula (1/2)⁽ⁿ⁻¹⁾, where n represents the term number.
The plot above shows the values of the sequence for n = 1, 2, 3, 4, 5. The values decrease by half with each successive term, illustrating exponential decay.
The horizontal axis (labeled “n”) represents the term number, while the vertical axis (labeled “Value”) represents the value of the term.
Exponential Growth Sequence: 2⁽ⁿ⁻¹⁾
This sequence is defined by the formula 2⁽ⁿ⁻¹⁾, where n represents the term number.
The plot above shows the values of the sequence for n = 1, 2, 3, 4, 5. The values double with each successive term, illustrating exponential growth.
The horizontal axis (labeled “n”) represents the term number, while the vertical axis (labeled “Value”) represents the value of the term.