Question

The ${11}^{th},{13}^{th}\mathrm{and}{15}^{th}$ terms of any G.P are in:

• A. G.P
• B. A.P
• C. H.P
• D. A.G.P

Answer 1

The correct option is A

G.P

Explanation for the correct option :

Step 1: Assumption

In Maths, Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio.

Let us consider the following G.P. with the initial term $a$ and the common ratio $r$ :

$a,ar,a{r}^2,a{r}^3,\cdots ,a{r}^n,\dots$

Step 2: Finding the ${11}^{th},{13}^{th}\mathrm{and}{15}^{th}$ terms

The general term of the above G.P. will be: $a_n=a{r}^{n-1}$.

Therefore, we can write:

$$\begin{gathered} a_{11}=a{r}^{10} \\ {a}_{13}=a{r}^{12} \\ {a}_{15}=a{r}^{14} \end{gathered}$$

Step 3: Checking whether these terms are in G.P.

Formula to be used: We know that the terms $a,b,c$ are in G.P. if $b^2=ac$.

Multiplying $a_{11}$and $a_{15}$, we get :

$$\begin{gathered} a_{11}{a}_{15}=a{r}^{10}a{r}^{14} \\ =a{r}^{24} \\ ={\left(a{r}^{12}\right)}^2 \\ ={\left(a_{13}\right)}^2 \end{gathered}$$

Thus, the terms $a_{11},a_{13}$ and $a_{15}$are in G.P.

Hence, option (A) is the correct answer.