Answer
414.9k+ views
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.
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)