Geometric Sequence Formula

Geometric Sequence Formula

The geometric sequence formula is used to describe a sequence in which each term is obtained by multiplying the previous term by the same constant value. This constant is called the common ratio. Geometric sequences are useful in mathematics, finance, science, and calculations involving repeated growth or decrease.

What Is a Geometric Sequence?

A geometric sequence is a list of numbers where the ratio between any two consecutive terms remains the same.

For example:

2, 6, 18, 54, 162, ...

Each term is multiplied by 3 to get the next term, so the common ratio is 3.

Geometric Sequence Formula

The formula for the nth term of a geometric sequence is:

a_n = a₁r^{n − 1}

Here, a_n is the nth term, a₁ is the first term, r is the common ratio, and n is the position of the term.

How to Find the Common Ratio

The common ratio can be found by dividing any term by the term immediately before it.

The formula is:

r = a_n / a_n−1

For example, in the sequence 5, 15, 45, 135:

r = 15 / 5 = 3

The ratio between each pair of consecutive terms is 3.

Example of the Geometric Sequence Formula

Suppose the first term is 4 and the common ratio is 2. To find the fifth term, use:

a₅ = 4(2)^{5 − 1}

a₅ = 4 × 2⁴

a₅ = 64

Therefore, the fifth term is 64.

Sum of a Geometric Sequence

The sum of the first n terms of a geometric sequence can be found with:

S_n = a₁(1 − rⁿ) / (1 − r)

This formula is used when r ≠ 1.

For example, consider the sequence:

2, 4, 8, 16

Here, a₁ = 2, r = 2, and n = 4.

S₄ = 2(1 − 2⁴) / (1 − 2)

S₄ = 30

So, 2 + 4 + 8 + 16 = 30.

Infinite Geometric Sequence Formula

If the absolute value of the common ratio is less than 1, an infinite geometric series has a finite sum.

The formula is:

S = a₁ / (1 − r)

This formula is valid when:

|r| < 1

For example, if the first term is 8 and the common ratio is 1/2:

S = 8 / (1 − 1/2) = 16

Geometric Sequence vs. Arithmetic Sequence

A geometric sequence changes by multiplying each term by a constant ratio. An arithmetic sequence changes by adding or subtracting the same constant difference from each term.

For example, 3, 6, 12, 24 is geometric because each term is multiplied by 2. In contrast, 3, 6, 9, 12 is arithmetic because 3 is added to each term.

Important Points to Remember

The first term and common ratio determine every term in a geometric sequence. A positive ratio produces terms with the same sign when the first term is positive, while a negative ratio causes the signs to alternate. A ratio greater than 1 generally produces increasing magnitudes, while a ratio between 0 and 1 produces decreasing magnitudes.

Geometric Sequence Formula in Short

The main formula for finding the nth term is:

a_n = a₁r^{n − 1}

For the sum of the first n terms:

S_n = a₁(1 − rⁿ) / (1 − r)

For an infinite geometric series with |r| < 1:

S = a₁ / (1 − r)

These formulas make it easier to find individual terms and calculate the sum of geometric sequences without adding every term separately.