The recursive formula describes an arithmetic sequence. ﻿ a_1 =12 𝑎 1 = 12 a 1 ​ =12﻿ ﻿ a_n = a_{n-1}+7 𝑎 𝑛 = 𝑎 𝑛 − 1 + 7 a n ​ =a n−1 ​ +7﻿. Find the sum of the first 15 terms in the sequence. 

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**.

---

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.

The recursive formula describes an arithmetic sequence. a_1 = 12, a_n = a_{n-1} + 7. Find the sum of the first 15 terms in the sequence.

Explore how to calculate the sum of the first 15 terms in an arithmetic sequence with a first term of 12 and a common difference of 7. This problem is ideal for students in Grades 9-10, covering key concepts in arithmetic sequences and series.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation