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# In a G.P. the ratio of the sum of the first eleven terms to the sum of last eleven terms is $\dfrac{1}{8}$ and the ratio of the sum of all terms without the first nine to the sum of all the terms without the last nine is $2$ .Then the number of terms of the G.P. is A) $15$ B) $43$ C) $38$ D) $56$

Last updated date: 18th Jun 2024
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Answer
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Hint: In this question, we have to find the number of terms in the geometric progression.
First we need to find out the equations from the ratios given by using the geometrical progression sum formula, then solving these two equations we will get the required solution.

Formula used: The sum of the G.P. series is represented by $a + ar + a{r^2} + a{r^3} + ...... + a{r^{n - 1}}$
$\sum\limits_{k = 0}^{n - 1} {a{r^k} = a\left( {\dfrac{{1 - {r^n}}}{{1 - r}}} \right)}$
Where ‘ a’ is the first term, and r is the factor between the terms (called the "common ratio") and n is the number of terms in the G.P.

Complete step-by-step answer:
It is given that, in a G.P. The ratio of the sum of the first eleven terms to the sum of last eleven terms is $\dfrac{1}{8}$.
We need to find out the number of terms of the G.P.
The sum of first eleven terms is $\sum\limits_{k = 0}^{10} {a{r^k} = a\left( {\dfrac{{1 - {r^{11}}}}{{1 - r}}} \right)}$
Sum of last eleven terms is $a{r^{n - 11}}\left( {\dfrac{{1 - {r^{11}}}}{{1 - r}}} \right)$
Using the sum formula for G.P. in the given ratio we get,
$\Rightarrow$$\dfrac{{a\left( {\dfrac{{1 - {r^{11}}}}{{1 - r}}} \right)}}{{a{r^{n - 11}}\left( {\dfrac{{1 - {r^{11}}}}{{1 - r}}} \right)}} = \dfrac{1}{8}$
${r^{n - 11}} = 8......(1)$
Also given that, the ratio of the sum of all terms without the first nine to the sum of all the terms without the last nine is $2$.
When we take the sum of all terms without the first nine then the first term is $a{r^9}$.
Then using the formula we get,
$\Rightarrow$$\dfrac{{a{r^9}\left( {\dfrac{{1 - {r^{n - 9}}}}{{1 - r}}} \right)}}{{a\left( {\dfrac{{1 - {r^{n - 9}}}}{{1 - r}}} \right)}} = 2$
By solving this we get,
$\Rightarrow$${r^9} = 2$
$\Rightarrow$$r = {2^{\dfrac{1}{9}}}......(2)$
Equation (1) is rewritten as follows,
$\Rightarrow$${r^{n - 11}} = 8 = {2^3}$
Let us substitute (2) in the above equation we get,
$\Rightarrow$${\left( {{2^{\dfrac{1}{9}}}} \right)^{n - 11}} = {2^3}$
$\Rightarrow$${2^{\dfrac{{n - 11}}{9}}} = {2^3}$
Let us now compare the powers of the above equation we get,
$\Rightarrow$$\dfrac{{n - 11}}{9} = 3$
$\Rightarrow$$n = 11 + 27 = 38$
Hence, the number of terms is 38.

Therefore (C) is the correct option.

Additional Information: A geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called a common ratio.
In General we write a Geometric Sequence like this $\left\{ {a,ar,a{r^2},a{r^3},.......} \right\}$, where ‘a’ is the first term, and r is the factor between the terms (called the "common ratio").

Note: Here we are provided with a ratio between two different sets of numbers in the geometric progression which has led us to find an equation in r, since there is another relation we will relate it so that we can get another clue regarding r. This is a basic idea used in solving most of the problems.