By Principle of Mathematical Induction :
nth term of A.P. whose first term is 'a' and common difference is 'd' is ______

Answer

To find the nth term of an arithmetic progression (A.P.) whose first term is \( a \) and common difference is \( d \), we will use the Principle of Mathematical Induction.

### Step-by-Step Solution:

1. **Understanding the nth term of A.P.:**
The nth term \( T_n \) of an A.P. can be expressed as:
\[ T_n = a + (n - 1) \cdot d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Base Case:**
We start by verifying the base case, \( n = 1 \):
\[ T_1 = a + (1 - 1) \cdot d = a + 0 = a \]
This shows that the formula holds true for \( n = 1 \).

3. **Inductive Hypothesis:**
Assume that the formula holds for \( n = k \):
\[ T_k = a + (k - 1) \cdot d \]
We will assume this is true for some arbitrary positive integer \( k \).

4. **Inductive Step:**
We need to prove that the formula holds for \( n = k + 1 \):
\[ T_{k+1} = a + (k + 1 - 1) \cdot d = a + k \cdot d \]
According to our inductive hypothesis, we know:
\[ T_k = a + (k - 1) \cdot d \]
The difference between the \( k+1 \)th term and the \( k \)th term is the common difference \( d \):
\[ T_{k+1} = T_k + d \]
Substituting the value of \( T_k \):
\[ T_{k+1} = \left( a + (k - 1) \cdot d \right) + d \]
Simplifying this expression:
\[ T_{k+1} = a + (k - 1) \cdot d + d = a + k \cdot d \]
Thus, we have shown that if the formula holds for \( n = k \), it also holds for \( n = k + 1 \).

5. **Conclusion:**
Since the base case is true and the inductive step has been proven, by the principle of mathematical induction, the formula for the nth term of an A.P. is:
\[ T_n = a + (n - 1) \cdot d \]