If the sum of the first n terms of an AP is given by (n2 + 5n + 7),determine its 15th term.

1. Break Down the Problem

To find the 15th term of the arithmetic progression (AP), we first need to determine the general formula for the $n$-th term of the AP. We are given that the sum of the first $n$ terms $S_n = n^2 + 5n + 7$. The $n$-th term can be expressed in terms of the sum of the first $n$ terms and the sum of the first $n-1$ terms.

2. Relevant Concepts

The formula for the $n$-th term $a_n$ of the AP can be derived from the sum of the first $n$ terms: $a_n = S_n - S_{n-1}$ Where $S_n$ is the sum of the first $n$ terms, and $S_{n-1}$ is the sum of the first $n-1$ terms.

3. Analysis and Detail

First, we calculate $S_{n-1}$: $S_{n-1} = (n-1)^2 + 5(n-1) + 7$

Expanding $S_{n-1}$: $S_{n-1} = (n^2 - 2n + 1) + (5n - 5) + 7$ $S_{n-1} = n^2 + 3n + 3$

Now, calculate $a_n$: $a_n = S_n - S_{n-1}$ $a_n = (n^2 + 5n + 7) - (n^2 + 3n + 3)$ $a_n = (n^2 + 5n + 7 - n^2 - 3n - 3)$ $a_n = 2n + 4$

4. Verify and Summarize

Now that we have $a_n = 2n + 4$, we can find the 15th term $a_{15}$: $a_{15} = 2(15) + 4 = 30 + 4 = 34$

Final Answer

The 15th term of the arithmetic progression is $34$.