To find the sum of the first 15 terms of an arithmetic sequence defined recursively, we start by identifying the first term $a_1$ and the common difference $d$.

Given:

• The first term $a_1 = 12$
• The recursive formula $a_n = a_{n-1} + 7$, which indicates that each term is 7 more than the previous term. Hence, the common difference $d = 7$.

Step 1: Find the 15th term $a_{15}$

The general formula for the $n$-th term of an arithmetic sequence is:

$a_n = a_1 + (n - 1) \cdot d$

Substitute $n = 15$, $a_1 = 12$, and $d = 7$:

$a_{15} = 12 + (15 - 1) \cdot 7$

$a_{15} = 12 + 14 \cdot 7$

$a_{15} = 12 + 98 = 110$

Step 2: Find the sum of the first 15 terms

The sum $S_n$ of the first $n$ terms of an arithmetic sequence is given by:

$S_n = \frac{n}{2} \cdot (a_1 + a_n)$

Substitute $n = 15$, $a_1 = 12$, and $a_{15} = 110$:

$S_{15} = \frac{15}{2} \cdot (12 + 110)$

$S_{15} = \frac{15}{2} \cdot 122$

$S_{15} = 15 \cdot 61$

$S_{15} = 915$

Final Answer

The sum of the first 15 terms in the sequence is 915.

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Would you like more details or have any questions? Here are some related questions to consider:

1. How is the recursive formula related to the arithmetic sequence?
2. What is the difference between arithmetic and geometric sequences?
3. How do you find the sum of the first $n$ terms for a geometric sequence?
4. How can we derive the formula for the $n$-th term in an arithmetic sequence?
5. What if the common difference was negative? How would it affect the sum?

Tip: For arithmetic sequences, knowing just the first term and the common difference allows you to find any term or sum easily.