If the third term of a G.P. is 42, then find the product of its first five terms

Options

A. (a) `42`
B. (b) `42^(5)`
C. (c) `98`
D. (d) `25^(5)`

Answer

To solve the problem step by step, we will follow the structure of a geometric progression (G.P.) and use the information given about the third term.

### Step 1: Understand the terms of a G.P.
In a geometric progression, the terms can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)
- Fifth term: \( ar^4 \)

### Step 2: Use the information given
According to the problem, the third term \( ar^2 \) is given as 42. Therefore, we can write:
\[ ar^2 = 42 \]

### Step 3: Find the product of the first five terms
The product of the first five terms of the G.P. can be expressed as:
\[ P = a \cdot ar \cdot ar^2 \cdot ar^3 \cdot ar^4 \]
This can be simplified:
\[ P = a^5 \cdot (r^0 \cdot r^1 \cdot r^2 \cdot r^3 \cdot r^4) = a^5 \cdot r^{0+1+2+3+4} = a^5 \cdot r^{10} \]

### Step 4: Express \( a^5 \cdot r^{10} \) in terms of \( ar^2 \)
Since we know \( ar^2 = 42 \), we can express \( a \) in terms of \( ar^2 \):
\[ a = \frac{ar^2}{r^2} = \frac{42}{r^2} \]
Now substituting \( a \) into the product:
\[ P = \left(\frac{42}{r^2}\right)^5 \cdot r^{10} \]
This simplifies to:
\[ P = \frac{42^5}{r^{10}} \cdot r^{10} = 42^5 \]

### Conclusion
Thus, the product of the first five terms of the G.P. is:
\[ P = 42^5 \]

### Final Answer
The product of the first five terms is \( 42^5 \).

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