Question

Find the 12 th term of a G.P. whose 8 th term is 192, and the common ratio is 2.

Answer 1

A Geometrical Progression is a sequence of numbers where each term is multiplied by its
previous number of sequences with a constant number known as a common ratio\[r\]. In general, a common ratio \[r\] is found by dividing any term of the series with its previous term. The behavior of a geometrical series depends on its common ratio.
If the ratio,
\[r = 1\]The progression is constant; all the terms in the series are the same.
\[r > 1\]The progression is increasing; all the subsequent terms in the series are increasing by the
common factor.
\[r < 1\], the progression is decreasing; all the subsequent terms in the series is decreasing by the
common factor.
Mathematically, a geometric progression series is summarized as
\[{a_1},{a_1}r,{a_1}{r^2},{a_1}{r^3}........\]where \[{a_1}\] is the first term of series and $r$ is the
common ratio. In general, for the nth term of a geometric series progression, ${t_n} = a{r^{\left( {n - 1}
\right)}}$ where a is the first term, r is the common ratio, and n is the number of terms.
Complete step by step solution:
For the 8 th term, substitute $n = 8$, ${t_8} = 192$ and for the common ratio, substitute $r = 2$ in the
formula ${t_n} = a{r^{\left( {n - 1} \right)}}$ to determine the first term of the geometric series
progression.
$
{t_n} = a{r^{\left( {n - 1} \right)}} \\
192 = a{\left( 2 \right)^{8 - 1}} \\
192 = a \times {2^7} \\
a = \dfrac{{192}}{{{2^7}}} = \dfrac{{192}}{{128}} = \dfrac{3}{2} \\
$
Now, to determine the value of the 12 th term of the geometric series progression, substitute $n = 12$,
and $r = 2$ in the formula ${t_n} = a{r^{\left( {n - 1} \right)}}$.
$
{t_n} = a{r^{\left( {n - 1} \right)}} \\
{t_{12}} = \dfrac{3}{2} \times {\left( 2 \right)^{12 - 1}} \\
= \dfrac{3}{2} \times {2^{11}} \\
= 3 \times {2^{10}} \\
= 3 \times 1024 \\
= 3072 \\
$
Note: Before proceeding with the calculations in the series question, we need to find the nature of the
series, i.e., A.P., G.P., or H.P. series. For the G.P. series, the common ratio is a critical factor as it
determines the nature of the series, whether increasing or decreasing. Alternatively, we can multiply the
common ratio to determine the result as well.