The third term of a G.P. is 3, the product of first five terms of this progression is:a)81b)27c)243d)none of theseCorrect answer is option 'C'. Can you explain this answer?

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9 is correct

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Can anyone please show me the workout of this problem!!

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Answer is option 'C'

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243Is the answer

Answer

Given Information:
The third term of the geometric progression (G.P.) is 3.

Objective:
To find the product of the first five terms of this progression.

Solution:

Formula for the nth term of a G.P.:
The nth term of a G.P. can be calculated using the formula: $a_n = a_1 \times r^{(n-1)}$, where:
- $a_n$ is the nth term of the G.P.
- $a_1$ is the first term of the G.P.
- $r$ is the common ratio of the G.P.
- $n$ is the position of the term in the G.P.

Given:
The third term of the G.P. is 3. Let the first term be $a_1$ and the common ratio be $r$.
So, the third term can be written as: $a_3 = a_1 \times r^{(3-1)} = a_1 \times r^2 = 3$

Product of the first five terms:
The product of the first five terms of the G.P. can be calculated as: $a_1 \times a_2 \times a_3 \times a_4 \times a_5 = a_1 \times (a_1 \times r) \times 3 \times (a_1 \times r^3) \times (a_1 \times r^4)$

Product Calculation:
Since we know that $a_3 = 3$, we can substitute this value in the product calculation:
$3 = a_1 \times r^2$
$a_1 = \frac{3}{r^2}$
Substitute $a_1$ back into the product calculation and simplify to find the product of the first five terms.

Final Answer:
The correct answer is option 'C' (243).

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