Question

The third term of a G.P. is the square of first term. If the second term is \[{8^{th}}\] then the \[{6^{th}}\] term is
A. 120
B. 124
C. 128
D. 132

Answer 1

Hint:
Here, in this problem we have been given that the third term of a G.P is the square of first term. We know that ‘a’ is the first term of G.P and r is the common ratio and then the G.P will be as follows\[a,ar,a{r^2},a{r^3},....\]. To find the 6^th term we have to use the general formula of G.P \[{T_r} = a{r^{n - 1}}\]. Then we have to substitute the values of “a” and “r” in the formula to determine the value of sixth term.
Formula use:
To calculate Geometric progression, we use the formula
\[a{r^{n - 1}}\]
Where: \[a\] is the first term and \[r\] is the common ratio.
Complete step-by-step solution
We have been given that the third term of the G.P., according to the question, is the square of first term.
Let \[a\] and \[r\] be the G.P.'s initial term and common ratio, respectively.
Let \[a,ar,a{r^2}\] in the GP.
Given in the data that,
\[a{r^2} = {a^2}\]--- (1)
\[ar = 8\]--- (2)
From the equation (1)
Now, we have to move \[a\]from right hand side of the equation to LHS’s denominator, we get
\[{r^2} = \frac{{{a^2}}}{a}\]
Now, from the above equation cancel the similar terms, we get
\[{r^2} = a\]-- (3)
We have been given that second term is\[ar = 8\]
Thus, we obtain as per formula
\[ \Rightarrow {r^3} = 8\]-- (4)
Let us solve the equation (3) and equation (4), we obtain
\[ \Rightarrow r = 2\]
And the value of \[a = 4\]
Since \[{6^{th}}\] term is given by \[a{r^5}\]
We have to write that as below,
\[{a_6} = a{r^5}\]
Now, we have to substitute the calculated value of \[a\] and \[r\]in the above formula, we get
\[ = 4{(2)^5}\]
Now, solve the power of \[2\]in the above equation
\[ = 4(32)\]
Multiply the terms, we get
\[{a_6} = 128\]
Therefore, the \[{6^{th}}\] term of Geometric progression is \[128\]
Hence, the option C is correct.
Note:
A geometric progression is a series in which any two consecutive words have the same ratio. A geometric series' common ratio can be found by dividing any two consecutive terms of the series. If we know the first term and the series' common ratio, we can calculate any term of a geometric progression. So, one should be thorough with the G.P formulas and their series to get the desired solution.