What is a geometric sequence and how to calculate it?

What is a geometric sequence and how to calculate it?

Sequences are widely used in mathematics and statistics to perform various tasks. The geometric sequence is one of the types of sequence used to measure the constant ratio to its previous term. This type of sequence can be finite or infinite.

The term that is constant or has a common ratio can be positive or negative. In this post, we will learn about the definition and formulas of geometric sequence with a lot of examples.

Geometric Sequence

A sequence that is obtained by multiplying the constant term to the successive terms is known as a geometric sequence. The constant term that is multiplied by each previous digit is known as the common ratio. If the sequence doesn’t have a common ratio is not a geometric sequence.

In simple words, multiply the common ratio to each previous number to form a sequence or to multiply the first term by the powers of common ratios starting from zero. The word geometric progression can also be written in place of a geometric sequence as both words are similar.

The constant value of the geometric sequence can have both negative as well as positive values. To find the terms of a geometric series, we only need the first term and the constant ratio. There is no common difference available in this sequence like the arithmetic sequence.

The geometric sequence can either be finite or infinite. If n terms are given the sequence is known as a finite geometric sequence. While if the infinite terms are given then the sequence is known as infinite.

Formulas of the geometric sequence

There are three formulas of the geometric sequence to find the sequence by using a common ratio and the other two are used to find the sum of the sequence for finite or infinite terms.

  • For n terms

Geometric sequence = x = x, xu, xu2, xu3, …

Geometric sequence = xn = xun-1

In the above formula, xn is the nth term of the sequence that we want to determine, x1 is the first term of the sequence, and u is the common ratio of the sequence. You can use the geometric sequence calculator to get the result of your question according to the formula.

  • For the sum of the finite terms

The sum of the finite geometric sequence = Sn = x (un – 1) / (u – 1)

In the above equation, x is the first term of the sequence.

  • For the sum of infinite terms.

Sum of the infinite geometric sequence = S = x/1 – u

How to calculate geometric sequence?

For the calculation of the geometric sequence, we can use formulas for this sequence.

Example 1: For the nth term

Find the 19th term of the geometric sequence if the first term of the sequence is 5 and the common ratio of the sequence is 3?

Solution

Step 1: Identify the given values.

First term = x = 5

Common ratio = u = 3

n = 19

Step 2: Now take the general formula to find the nth term of the geometric sequence.

Geometric sequence = xn = xun-1

Step 3: Put the given data in the above formula.

Geometric sequence = x19 = 5 * 319-1

Geometric sequence = x19 = 5 * 318

Geometric sequence = x19 = 5 * (3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3)

Geometric sequence = x19 = 5 * (387420489)

Geometric sequence = x19 = 1937102445

Hence, the 19th term of the sequence is 1937102445.

Example 2: For the sum of a finite sequence

Find the sum of the first 5 terms of the geometric sequence if the first term of the sequence is 15 and the common ratio of the sequence is 13?

Solution

Step 1: Identify the given values.

First term = x = 15

Common ratio = u = 13

n = 5

Step 2: Now take the general formula to find the sum of finite terms of the geometric sequence.

Sum of the finite geometric sequence = Sn = x (un – 1) / (u – 1)

Step 3: Put the given data in the above formula.

Sum of the finite geometric sequence = S5 = 15(135 – 1) / (13 – 1)

Sum of the finite geometric sequence = S5 = 15(135 – 1) / (12)

Sum of the finite geometric sequence = S5 = 15(371293 – 1) / (12)

Sum of the finite geometric sequence = S5 = 15(371292) / (12)

Sum of the finite geometric sequence = S5 = 1856460 / (12)

Sum of the finite geometric sequence = S5 = 154705

Example 3:

Find the sum of the first 23 terms of the geometric sequence if the first term of the sequence is 4 and the common ratio of the sequence is 2?

Solution

Step 1: Identify the given values.

First term = x = 4

Common ratio = u = 2

n = 23

Step 2: Now take the general formula to find the sum of finite terms of the geometric sequence.

Sum of the finite geometric sequence = Sn = x (un – 1) / (u – 1)

Step 3: Put the given data in the above formula.

Sum of the finite geometric sequence = S23 = 4(223 – 1) / (23 – 1)

Sum of the finite geometric sequence = S23 = 4(223 – 1) / (22)

Sum of the finite geometric sequence = S23 = 4(8388608 – 1) / (22)

Sum of the finite geometric sequence = S23 = 4(8388607) / (22)

Sum of the finite geometric sequence = S23 = 33554428 / (22)

Sum of the finite geometric sequence = S23 = 1525201.273

Wrapping up

A geometric sequence is very essential to perform certain tasks and problems of mathematics and statistics. Use the formulas discussed above to find the geometric sequence and other related terms.

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