The terms of a G.P. with first term a and common ratio r are squared. Is the resulting sequence also a G.P.? If its is so, find the its first term, common ratio and the nth term.

Answer

To determine whether the sequence obtained by squaring the terms of a geometric progression (G.P.) is also a G.P., we start with the definition of a G.P.

### Step 1: Identify the terms of the original G.P.
The terms of a G.P. can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)
- And so on...

The general term of the G.P. can be represented as:
\[ T_n = ar^{n-1} \]

### Step 2: Square the terms of the G.P.
Now, we square each term of the G.P.:
- First term: \( a^2 \)
- Second term: \( (ar)^2 = a^2 r^2 \)
- Third term: \( (ar^2)^2 = a^2 r^4 \)
- Fourth term: \( (ar^3)^2 = a^2 r^6 \)
- And so on...

The squared terms can be expressed as:
\[ T_n' = (ar^{n-1})^2 = a^2 r^{2(n-1)} = a^2 r^{2n-2} \]

### Step 3: Check if the squared terms form a G.P.
To check if the squared terms form a G.P., we need to find the common ratio of the new sequence. The first term of the new sequence is:
\[ T_1' = a^2 \]
The second term is:
\[ T_2' = a^2 r^2 \]
The common ratio \( r' \) can be calculated as:
\[ r' = \frac{T_2'}{T_1'} = \frac{a^2 r^2}{a^2} = r^2 \]

### Step 4: Write the general term of the new G.P.
The general term of the new sequence (squared terms) can be written as:
\[ T_n' = a^2 r^{2n-2} \]

### Conclusion
Thus, the resulting sequence is indeed a G.P. with:
- First term: \( a^2 \)
- Common ratio: \( r^2 \)
- The nth term: \( T_n' = a^2 r^{2n-2} \)

### Summary of Results:
- First term: \( a^2 \)
- Common ratio: \( r^2 \)
- nth term: \( a^2 r^{2n-2} \)