# The ${{\text{n}}^{{\text{th}}}}$ term of a geometric progression is ${{\text{a}}_{\text{n}}}$= ${\text{a}}{{\text{r}}^{{\text{n - 1}}}}$, where r represents

A. Common difference

B. Common ratio

C. First term

D. Radius

Last updated date: 01st Apr 2023

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Answer

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Hint: Geometric progression is a sequence in which each term is multiplied by a common factor to obtain the next term.

Complete step-by-step answer:

Given, ${{\text{n}}^{{\text{th}}}}$term of a geometric progression is ${{\text{a}}_{\text{n}}}$, and it is equal to ${\text{a}}{{\text{r}}^{{\text{n - 1}}}}$. We need to find what r represents.

The geometric progression is a progression of numbers with a constant ratio between each number and the one before. If the first term is k and the common ratio is m, then the geometric progression will be k, km, km$^2$, km$^3$,…, km$^{{\text{n - 1}}}$. Here , the nth term is km$^{{\text{n - 1}}}$. Comparing it with ${\text{a}}{{\text{r}}^{{\text{n - 1}}}}$, we get k = a and m = r i.e. a is the first term of the geometric progression and r is the common ratio.

Hence, option (B) is correct.

Note:-We generally have three types of progression. Arithmetic progression, Geometric progression and harmonic progression. The e.g. of geometric progression is 1,3,9,27… .In this example the first term is 1 and the common ratio is 3.

Complete step-by-step answer:

Given, ${{\text{n}}^{{\text{th}}}}$term of a geometric progression is ${{\text{a}}_{\text{n}}}$, and it is equal to ${\text{a}}{{\text{r}}^{{\text{n - 1}}}}$. We need to find what r represents.

The geometric progression is a progression of numbers with a constant ratio between each number and the one before. If the first term is k and the common ratio is m, then the geometric progression will be k, km, km$^2$, km$^3$,…, km$^{{\text{n - 1}}}$. Here , the nth term is km$^{{\text{n - 1}}}$. Comparing it with ${\text{a}}{{\text{r}}^{{\text{n - 1}}}}$, we get k = a and m = r i.e. a is the first term of the geometric progression and r is the common ratio.

Hence, option (B) is correct.

Note:-We generally have three types of progression. Arithmetic progression, Geometric progression and harmonic progression. The e.g. of geometric progression is 1,3,9,27… .In this example the first term is 1 and the common ratio is 3.

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