Geometric sequence formulas and examples

A sequence of numbers where each term after the first is got by multiplying the previous one by a fixed number. That number is called the common ratio.

For example, the sequence 2, 6, 18, 54, 162, 486, 1458, . . . is a geometric progression with a common ratio of 3.

Rule of geometric sequence

General rule of geometric sequence can be written by this formula, where $b_1$ - the first term, $q$ - the factor between the terms, $n$ - the number of term.

$$b_n=b_1\cdot q^{n-1}$$

Write a rule, and calculate the 10th term, for this geometricic sequence: 3, 6, 12, 24, 48...

The first term value of this sequence is $b_1=3$. This sequence has a factor of 2 between each number, so $q=2$. Using the geometric sequence rule we get:

$$b_n=b_1\cdot q^{n-1}=3\cdot 2^{n-1}$$

The 10th term of this geometric sequence can be find easily using expression above:

$$b_{10}=3\cdot 2^{10-1}=1536$$

Let we have geometric sequence where $b_1=2$ and $q=3$. What is the number of term with value 4374?

Using the geometric sequence rule we can write simple equation and solve it

\begin{align} 4374 & =2\cdot 3^{n-1} \\ 2187 & = 3^{n-1} \\ 3^7 & = 3^{n-1} \\ n & =8 \end{align}

Sum of geometric sequence

To sum up the terms of a geometric sequence is simple only when we have not many terms. But for sequences that have large number of terms we can use this formula below.

$$\displaystyle S_n=\frac{b_1(q^n-1)}{q-1}$$

Let we have geometric sequence where $b_1=4$ and $q=2$. What is the sum of 10 first terms in this sequence?

Using the geometric sequence sum formula we get:

$$\displaystyle S_{10}=\frac{4(2^{10}-1)}{2-1}=4092$$

2021-07-29