If the sum of n terms of an AP is `2n^(2) + 7n`, then which of its terms is 113?

Answer

To solve the problem step by step, we will first find the nth term of the arithmetic progression (AP) using the given sum of n terms, and then determine which term is equal to 113.

### Step 1: Write down the formula for the sum of n terms
The sum of the first n terms of the AP is given by:
\[ S_n = 2n^2 + 7n \]

### Step 2: Find the sum of the first (n-1) terms
To find the (n-1)th term, we need to calculate \( S_{n-1} \):
\[ S_{n-1} = 2(n-1)^2 + 7(n-1) \]
Expanding this:
\[ S_{n-1} = 2(n^2 - 2n + 1) + 7(n - 1) \]
\[ = 2n^2 - 4n + 2 + 7n - 7 \]
Combining like terms:
\[ S_{n-1} = 2n^2 + 3n - 5 \]

### Step 3: Find the nth term using the formula
The nth term \( T_n \) can be found using the formula:
\[ T_n = S_n - S_{n-1} \]
Substituting the values we have:
\[ T_n = (2n^2 + 7n) - (2n^2 + 3n - 5) \]
Simplifying this:
\[ T_n = 2n^2 + 7n - 2n^2 - 3n + 5 \]
\[ = 4n + 5 \]

### Step 4: Set the nth term equal to 113
We are given that one of the terms is 113, so we set up the equation:
\[ 4n + 5 = 113 \]

### Step 5: Solve for n
Subtract 5 from both sides:
\[ 4n = 113 - 5 \]
\[ 4n = 108 \]
Now, divide by 4:
\[ n = \frac{108}{4} \]
\[ n = 27 \]

### Conclusion
Thus, the 27th term of the AP is 113.

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