Question

Write the G.P, if the first term is $ a = 3 $ and the common ratio is $ r = 2 $ .

Answer 1

Hint: In order to find the G.P series, we should first know what exactly G.P is. G.P stands for Geometric Progression. It’s a series in which the next term is the product of the previous term and the common ratio. Find the next four, five terms using the first term and the common ratio given and the G.P series is obtained.

Complete step by step solution:
We are given the first term of a G.P series which is $ a = 3 $ and the common ratio of the series which is $ r = 2 $ .
Since, we know that in G.P the next term of the previous number is calculated by multiplying the common ratio with the previous term.
 $ {T_1} = $ First term $ \left( a \right) = 3 $
According to the definition, the next term would be $ {T_2} = $ $ ar = 3 \times 2 = 6 $ .
Similarly, the next term would be the ratio multiplied by the previous number $ ar $ . So, the third term becomes:
 $ {T_3} = \left( {ar} \right)r = a{r^2} $
Substituting the values, we get:
 $ {T_3} = a{r^2} = \left( 3 \right){\left( 2 \right)^2} = 3 \times 4 = 12 $
Similarly, the series would go on:
So, the next term is:
 $ {T_4} = \left( {a{r^2}} \right)r = a{r^3} $
 $ {T_4} = a{r^3} = \left( 3 \right){\left( 2 \right)^3} = 3 \times 8 = 24 $
Similarly, the process would continue till the number of terms we want to find the series.
Therefore, the general G.P series we obtained is: $ a,ar,a{r^2},a{r^3},....... $
And, hence the G.P series for the first term is $ a = 3 $ and the common ratio is $ r = 2 $ : $ 3,6,12,24,....... $ .
So, the correct answer is “$ 3,6,12,24,....... $”.

Note: We can find the G.P series till any number of terms we want using the same method, and following the same steps.
If we are given to find the ‘nth term instead of finding the whole series, we would use the formula $ {T_n} = {a_1}{r^{\left( {n - 1} \right)}} $ , substitute the values and get the result.