In an A.P., if common difference is 2, sum to n terms is 49, 7th term is 13, then n=

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem step by step, we will use the properties of an arithmetic progression (A.P.).

### Given:
- Common difference (d) = 2
- Sum of n terms (S_n) = 49
- 7th term (a_7) = 13

### Step 1: Find the first term (a)
The formula for the nth term of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]

For the 7th term (n = 7):
\[ a_7 = a + (7 - 1) \cdot d \]
\[ 13 = a + 6 \cdot 2 \]
\[ 13 = a + 12 \]

Now, solving for a:
\[ a = 13 - 12 \]
\[ a = 1 \]

### Step 2: Use the sum formula for n terms
The formula for the sum of the first n terms of an A.P. is:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d) \]

Substituting the known values:
\[ 49 = \frac{n}{2} \cdot (2 \cdot 1 + (n - 1) \cdot 2) \]
\[ 49 = \frac{n}{2} \cdot (2 + 2(n - 1)) \]
\[ 49 = \frac{n}{2} \cdot (2 + 2n - 2) \]
\[ 49 = \frac{n}{2} \cdot (2n) \]
\[ 49 = n^2 \]

### Step 3: Solve for n
To find n, we rearrange the equation:
\[ n^2 = 49 \]
Taking the square root of both sides:
\[ n = 7 \]

### Conclusion
Thus, the value of n is:
\[ n = 7 \]

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