If for every `n in N`, sum to n terms of an A.P. is `5n^(2)+7n` then is 10th term is

Options

A. `7//2`
B. 570
C. 102
D. 52

Answer

To find the 10th term of the arithmetic progression (A.P.) where the sum of the first n terms is given by \( S_n = 5n^2 + 7n \), we can follow these steps:

### Step 1: Understand the formula for the sum of n terms of an A.P.
The sum of the first n terms of an A.P. can be expressed as:
\[ S_n = \frac{n}{2} \left(2a + (n - 1)d\right) \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Set the given sum equal to the A.P. sum formula
We know from the problem that:
\[ S_n = 5n^2 + 7n \]
Equating the two expressions gives:
\[ \frac{n}{2} \left(2a + (n - 1)d\right) = 5n^2 + 7n \]

### Step 3: Simplify the equation
Multiplying both sides by 2 to eliminate the fraction:
\[ n(2a + (n - 1)d) = 10n^2 + 14n \]
Dividing both sides by \( n \) (assuming \( n \neq 0 \)):
\[ 2a + (n - 1)d = 10n + 14 \]

### Step 4: Rearrange the equation
Now, we can express this as:
\[ 2a + nd - d = 10n + 14 \]
Rearranging gives:
\[ nd + 2a - d = 10n + 14 \]

### Step 5: Compare coefficients
To find \( a \) and \( d \), we can compare coefficients of \( n \) and the constant terms:
- Coefficient of \( n \): \( d = 10 \)
- Constant term: \( 2a - d = 14 \)

### Step 6: Solve for \( a \)
Substituting \( d = 10 \) into the constant term equation:
\[ 2a - 10 = 14 \]
Adding 10 to both sides:
\[ 2a = 24 \]
Dividing by 2:
\[ a = 12 \]

### Step 7: Find the 10th term
The formula for the nth term of an A.P. is:
\[ a_n = a + (n - 1)d \]
To find the 10th term \( a_{10} \):
\[ a_{10} = 12 + (10 - 1) \cdot 10 \]
Calculating:
\[ a_{10} = 12 + 9 \cdot 10 = 12 + 90 = 102 \]

### Final Answer
Thus, the 10th term of the A.P. is:
\[ \boxed{102} \]