Infinite geometric sequence sum formula and examples

If we have geometric sequence, where $-1 < q < 1$, $q\neq 0$ and number of terms goes to infinity, it is called infinite geometric serie.

For example, the sequence $\small{16, 8, 4, 2, 1, 0.5, 0.25,...}$ is a infinite geometric progression with a common ratio of $\frac{1}{2}$.

The sum of all terms of infinite geometric sequence can be calculated by formula below, , where $b_1$ is the first term and $q$ is the factor between the terms.

$$S=\frac{b_1}{1-q}$$

Solved examples

What is the sum of all terms in this sequence: $\small{4, 2, 1, 0.5, 0.25...}$ ?

The first term value of this sequence is $b_1=4$. This sequence has a factor of 0.5 between each number, so $q=0.5$. Using the infinite geometric sequence sum formula we get:

$$S=\frac{4}{1-0.5}=8$$

Solve this equation $\sqrt{x}\cdot\sqrt[4]{x}\cdot\sqrt[8]{x}\cdot\sqrt[16]{x}...=10$

First of all we rewrite radicals in exponential form and using laws of exponents we get:

\begin{align} x^{\frac{1}{2}}\cdot x^{\frac{1}{4}}\cdot x^{\frac{1}{8}}\cdot x^{\frac{1}{16}}... & =10 \\ x^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}...} & =10 \end{align}

As we can see above the exponent of x is infinite geometric sequence which can be easily calculated with sum formula.

$$b_1=\frac{1}{2}, \ q=\frac{1}{2}, \ S=\frac{\frac{1}{2}}{1-\frac{1}{2}}=1$$

Answer: $x = 10$

Write this number $\small{0.27777...}$ as fraction.

$$0.27777...=0.2+0.07+0.007+0.0007+0.00007...$$ $$\frac{2}{10}+\frac{0.07}{1-0.1}=\frac{1}{5}+\frac{7}{90}=\frac{25}{90}=\frac{5}{18}$$

2021-07-29