Question

In a Geometric Progression, 3\(^{rd}\) term is 12, and 6\(^{th}\) term is 96. Find sum of first 5 terms.

• A. 93
• B. 186
• C. 248
• D. 124

Hint:

In GP problems, if you are given two terms, dividing the term with the higher index by the one with the lower index is a quick way to eliminate 'a' and find 'r'.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the 3rd and 6th terms of a Geometric Progression (GP). We need to find the sum of the first 5 terms (S\(_{5}\)).
Step 2: Key Formula or Approach:
The formula for the n\(^{th}\) term of a GP is a\(_{n}\) = ar\(^{n-1}\), where 'a' is the first term and 'r' is the common ratio.
The formula for the sum of the first n terms of a GP is S\(_{n}\) = \(\frac{a(r^n - 1)}{r - 1}\).
Step 3: Detailed Explanation:
We are given:
3rd term, a\(_{3}\) = ar\(^{3-1}\) = ar\(^{2}\) = 12 \quad --- (1)
6th term, a\(_{6}\) = ar\(^{6-1}\) = ar\(^{5}\) = 96 \quad --- (2)
To find the common ratio 'r', we divide equation (2) by equation (1):
\[ \frac{ar^5}{ar^2} = \frac{96}{12} \] \[ r^3 = 8 \] \[ r = \sqrt[3]{8} = 2 \] Now, substitute the value of r = 2 back into equation (1) to find the first term 'a':
\[ a(2)^2 = 12 \] \[ 4a = 12 \] \[ a = 3 \] Now we have the first term (a = 3) and the common ratio (r = 2). We can find the sum of the first 5 terms, S\(_{5}\):
\[ S_5 = \frac{a(r^5 - 1)}{r - 1} \] \[ S_5 = \frac{3(2^5 - 1)}{2 - 1} \] \[ S_5 = \frac{3(32 - 1)}{1} \] \[ S_5 = 3(31) = 93 \] Step 4: Final Answer:
The sum of the first 5 terms of the Geometric Progression is 93.