Question

If in a geometric progression \[\left\{ {{a_n}} \right\}\] , \[{a_1} = 3\] , \[{a_n} = 96\] and \[{S_n} = 189\] then the value of \[n\] is
(A) \[5\]
(B) \[6\]
(C) \[7\]
(D) \[8\]

Answer 1

Hint: In this question we are given the value of \[{a_1}\] and \[{a_n}\]. Substituting these values in \[a{r^{n - 1}} = {a_n}\] which is the formula for nth term of a gp, we can obtain the value of \[{r^n}\]. By substituting the value of a which is given and the value of \[{r^n}\]obtained above in the formula for \[{S_n}\] which is \[{S_n} = \dfrac{{a({r^n} - 1)}}{{(r - 1)}}\] we can obtain the value of r and hence the value of n can be found.

Complete step by step solution: 
Given the first term \[{a_{_1}}\] is
 \[{a_{_1}} = 3\]and \[{a_n} = 96\]
Therefore, the general \[{n^{th}}\] term of Geometric Progression is given as,
\[a{r^{n - 1}} = {a_n}\]
Here \[a{\text{ }} = {\text{ }}{a_1}{\text{ }} = {\text{ }}3\]and \[{a_n} = 96\]. Hence, we have
\[3{\text{ (}}{{\text{r}}^{n - 1}}){\text{ = }}96\]
This implies that,
\[{{\text{r}}^{n - 1}} = \dfrac{{96}}{3}\]
\[{{\text{r}}^{n - 1}} = 32\]
\[{r^{n - 1}}\] can be written as \[\dfrac{{{r^n}}}{r}\].
This implies that \[\dfrac{{{r^n}}}{r} = 32\]
 \[{r^n} = 32r\]… \[\left( 1 \right)\]
Since \[{S_n} = \dfrac{{a({r^n} - 1)}}{{(r - 1)}}\] we have,
\[{S_n} = \dfrac{{a({r^n} - 1)}}{{(r - 1)}} = 189\]
Since \[a = 3\] and \[{r^n} = 32r\] we have,
\[\dfrac{{3(32r - 1)}}{{(r - 1)}} = 189\]
\[\dfrac{{(32r - 1)}}{{(r - 1)}} = 63\]
Taking the denominator to the other side and simplifying we get,
\[32r - 1 = 63r - 63\]
\[32r - 63r = - 63 + 1\]
This implies that \[31r = 62\].
Hence \[r = 2\]
Substituting the value of r in equation \[\left( 1 \right)\] we get,
\[{2^n} = 32(2) = 64\]
Since \[64 = {2^6}\] we have,
\[{2^n} = {2^6}\]
Therefore, \[n = 6\]
Hence option (B) is correct.

Note: To obtain \[{r^n}\] from \[{r^{n - 1}}\], we can split the power of \[r\] i.e., \[n - 1\] as \[{r^n}.{r^{ - 1}}\] which is equal to \[\dfrac{{{r^n}}}{r}\]. By taking the \[r\] in the denominator to the other side, the value of \[{r^n}\] can be found. Also, it is important to convert 64 to a power of 2 so that on comparing with \[{2^n}\] the value of n is obtained.