Question

The sum of first four terms of a G.P. is $160$ and the common ratio is $3$, then the $4^{\text{th}}$ term is

• A. $118$
• B. $100$
• C. $108$
• D. $102$

Hint:

Once the first term is known, any term of a G.P. can be found using $a r^{n-1}$.

Answer 1

The Correct Option is C

Step 1: Writing the formula for sum of first $n$ terms of a G.P.
\[ S_n = a\frac{r^n - 1}{r - 1}, \quad r \neq 1 \]
Step 2: Substituting the given values.
Here, $S_4 = 160$ and $r = 3$. \[ 160 = a\frac{3^4 - 1}{3 - 1} = a\frac{81 - 1}{2} = 40a \]
Step 3: Finding the first term.
\[ a = \frac{160}{40} = 4 \]
Step 4: Finding the fourth term.
The fourth term is \[ a r^3 = 4 \times 3^3 = 4 \times 27 = 108 \]
Step 5: Conclusion.
The fourth term of the G.P. is $108$.