Is it true to say that `a_n=``3n^(2)` + 5 is the `n^(th)` term of an A.P. sequence? Justify your answer.

Answer

To determine whether the expression \( a_n = 3n^2 + 5 \) represents the \( n^{th} \) term of an arithmetic progression (A.P.), we need to check if the common difference between consecutive terms is constant.

### Step-by-Step Solution:

1. **Define the nth term**: We have \( a_n = 3n^2 + 5 \).

2. **Calculate the first few terms**:
- For \( n = 1 \):
\[ a_1 = 3(1^2) + 5 = 3 \cdot 1 + 5 = 8 \]
- For \( n = 2 \):
\[ a_2 = 3(2^2) + 5 = 3 \cdot 4 + 5 = 12 + 5 = 17 \]
- For \( n = 3 \):
\[ a_3 = 3(3^2) + 5 = 3 \cdot 9 + 5 = 27 + 5 = 32 \]
- For \( n = 4 \):
\[ a_4 = 3(4^2) + 5 = 3 \cdot 16 + 5 = 48 + 5 = 53 \]
- For \( n = 5 \):
\[ a_5 = 3(5^2) + 5 = 3 \cdot 25 + 5 = 75 + 5 = 80 \]

3. **List the calculated terms**: The first five terms are:
- \( a_1 = 8 \)
- \( a_2 = 17 \)
- \( a_3 = 32 \)
- \( a_4 = 53 \)
- \( a_5 = 80 \)

4. **Calculate the common differences**:
- \( d_1 = a_2 - a_1 = 17 - 8 = 9 \)
- \( d_2 = a_3 - a_2 = 32 - 17 = 15 \)
- \( d_3 = a_4 - a_3 = 53 - 32 = 21 \)
- \( d_4 = a_5 - a_4 = 80 - 53 = 27 \)

5. **Check if the common differences are constant**:
- We have \( d_1 = 9 \), \( d_2 = 15 \), \( d_3 = 21 \), and \( d_4 = 27 \).
- Since \( d_1 \neq d_2 \neq d_3 \neq d_4 \), the common differences are not constant.

6. **Conclusion**: Since the common difference is not the same for all terms, the sequence defined by \( a_n = 3n^2 + 5 \) is not an arithmetic progression.

### Final Answer:
The expression \( a_n = 3n^2 + 5 \) does not represent the \( n^{th} \) term of an arithmetic progression.

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