Finding Geometric Sequence explained with examples

A geometric sequence is a type of sequence. A sequence is generally the set of numbers having equal differences among the numbers and are separated by commas. The terms are used to create a common difference by multiplying the previous digit.

In this post, we will learn about the sequence, geometric sequence, and how to calculate it, with a lot of examples.

What is a Geometric sequence?

A geometric sequence usually states that a series of numbers comes by taking a common difference or by multiplying each term by a constant. The sequence is usually represented by the series but a little bit different in that the sequence is separated by commas while in series sum notation is used instead of commas.

A sequence is usually of the form like 1, 4, 16, 64, 256, 1024,… in this sequence all the terms are multiplied by themselves by 4 for making a common difference.

Formula of the geometric sequence

For geometric sequence, we use an equation to calculate the n^th term of the sequence having a constant difference.

b_n = b₁ r^n-1

In the above equation, b_n is the nth term of the sequence that we wish to calculate, b₁ is the first term of the sequence, r is the common difference or ratio between the terms, and n is the nth term of the sequence.

There are some other formulas of the geometric sequence used to calculate the sum of the sequence either for finite sequence or infinite sequence.

The formula used for the sum of the sequence for finite terms is given as.

Sum of the sequence for finite terms = b_n = b (1 – rⁿ) / 1 – r

Sum of the sequence for infinite terms = a / 1 – r

How to calculate the geometric sequence?

The geometric sequence can be calculated to find the nth term of the sequence. To apply the formula of the geometric sequence you must be familiar with the exponent. Let’s take some examples to calculate the geometric sequence to determine the nth term.

Example 1

Calculate the first 9 terms of the sequence if the initial value is 2, the common difference is 6.

Solution

Step 1: Determine the given values.

Initial term = 2

Total terms = 9

Total difference = 6

Step 2: Now take the values of n.

n = 1, 2, 3, 4, 5, 6, 7, 8, 9

Step 3: Take the general formula to calculate geometric sequence.

b_n = b₁ * r^n-1

Step 4: Now put the values of n one by one to find the first nine terms of the sequence.

Put n = 1

b_n = 2 * 6^n-1

b₁ = 2 * 6^1-1

b₁ = 2 * 6⁰

b₁ = 2 * 1 = 2

Put n = 2

b_n = 2 * 6^n-1

b₂ = 2 * 6^2-1

b₂ = 2 * 6¹

b₂ = 2 * 6

b₂ = 12

Put n = 3

b_n = 2 * 6^n-1

b₃ = 2 * 6^3-1

b₃ = 2 * 6²

b₃ = 2 * 36

b₃ = 72

Put n = 4

b_n = 2 * 6^n-1

b₄ = 2 * 6^4-1

b₄ = 2 * 6³

b₄ = 2 * 6 x 6 x 6

b₄ = 12 * 36

b₄ = 432

Put n = 5

b_n = 2 * 6^n-1

b₅ = 2 * 6^5-1

b₅ = 2 * 6⁴

b₅ = 2 * 6 x 6 x 6 x 6

b₅ = 2 * 1296

b₅ = 2592

Put n = 6

b_n = 2 * 6^n-1

b₆ = 2 * 6^6-1

b₆ = 2 * 6⁵

b₆ = 2 * 6 x 6 x 6 x 6 x 6

b₆ = 2 * 7776

b₆ = 15552

Put n = 7

b_n = 2 * 6^n-1

b₇ = 2 * 6^7-1

b₇ = 2 * 6⁶

b₇ = 2 * 6 x 6 x 6 x 6 x 6 x 6

b₇ = 2 * 46656

b₇ = 93312

Put n = 8

b_n = 2 * 6^n-1

b₈ = 2 * 6^8-1

b₈ = 2 * 6⁷

b₈ = 2 * 6 x 6 x 6 x 6 x 6 x 6 x 6

b₈ = 2 * 279936

b₈ = 559872

Put n = 9

b_n = 2 * 6^n-1

b₉ = 2 * 6^9-1

b₉ = 2 * 6⁸

b₉ = 2 * 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6

b₉ = 2 * 1679616

b₉ = 3359232

Step 5: Now write the first nine terms of the sequence separate them by commas.

2, 12, 72, 432, 2592, 15552, 93312, 559872, 3359232

For finding the nth term of the sequence, you can use a geometric sequence calculator instead of solving the problem with such large calculations.

Example 2

Find the sum of the sequence of the given terms, 1, 4, 16, 64, 256, 1024.

Solution

Step 1: Identify the values form the given terms.

n = 6

r = 4

b₁ = 1

Step 2: Take the general formula for the sum of the sequence.

Sum of the sequence for finite terms = b_n = b (1 – rⁿ) / 1 – r

Step 3: Put the identified values in the formula.

Sum of the sequence for finite terms = b_n = b (1 – rⁿ) / 1 – r

Sum of the sequence for finite terms = b₆ = 1 (1 – 4⁶) / 1 – 4

Sum of the sequence for finite terms = b₆ = (1 – 4 x 4 x 4 x 4 x 4 x 4) / -3

Sum of the sequence for finite terms = b₆ = (1 – 4096) / -3

Sum of the sequence for finite terms = b₆ = –4095/-3 = 4095/3

Sum of the sequence for finite terms = b₆ = 1365

Summary

The geometric sequence is very useful for finding the nth term of the sequence or the sum of the sequence for finite or infinite. This sequence is very essential for the sequences of the terms. By using formulas of this sequence, you can easily solve any problem of this sequence.

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