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How do I find the first term of a geometric sequence?

Last updated date: 13th Jun 2024
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Hint: In order to determine the first term of the geometric sequence, we will consider a general geometric sequence and write in the form of the sum of the terms. Then, we will multiply $\left( {1 - r} \right)$ to both the sides of the equation. By which all the other terms will be cancelled. And we will get the required formula.

Complete step by step solution:
In general a geometric sequence can be written as,
$a,ar,a{r^2},a{r^3},a{r^4} \ldots a{r^{n - 1}}$
Where $a$ is the first term and $r$ is the factor between the terms.
The sum of the terms is,
$\sum\limits_{k = 1}^n {a{r^{k - 1}} = } a + ar + a{r^2} + a{r^3} + a{r^4} \ldots a{r^{n - 1}}$
Now, let us multiply $\left( {1 - r} \right)$ to both the sides of the equation, we have,
$\left( {1 - r} \right)\sum\limits_{k = 1}^n {a{r^{k - 1}} = } \left( {1 - r} \right)a + ar + a{r^2} + a{r^3} + a{r^4} \ldots a{r^{n - 1}}$
$\left( {1 - r} \right)\sum\limits_{k = 1}^n {a{r^{k - 1}} = } a + ar + a{r^2} + a{r^3} + a{r^4} \ldots a{r^{n - 1}} - ar - a{r^2} - a{r^3} - a{r^4} \ldots a{r^{n - 1}} - a{r^n}$
Since, all the other terms cancel.
If $r \ne 1$, we can rearrange the above equation to get convenient formula for a geometric series that computes the sum of $n$ terms,
$\sum\limits_{k = 1}^n {a{r^{k - 1}} = } \dfrac{{a\left( {1 - {r^n}} \right)}}{{\left( {1 - r} \right)}}$
If the ${n^{th}}$ term of the sequence are given, then we can use,
${a_n} = a{r^{n - 1}}$

Note: A sequence is a set of things (usually numbers) that are in order. In a geometric sequence each term is found by multiplying the previous term by a constant. Geometric progression is also known as geometric sequence.

Geometric series are used throughout mathematics. They have important applications in physics, engineering, biology, economics, computer science, Queuing theory, and finance. Geometric series are one of the simplest examples of the infinite series with finite sums, although not all of them have this property. Geometric growth is found in many real life scenarios such as population growth and the growth of an investment. Geometric series often arise as the perimeter, area or volume of a self-similar figure.