https://file.aiquickdraw.com/mathbot/file/1727529135435wmx1034e.jpg

The image contains a mathematical problem in Gujarati. Here is the transcription and breakdown:

**Transcription:**

> \( a_n = 9 - 5n \)
> 
> Shodhviye shreni nu pratham 15 padhono sarvallo shodhavo.

**Translation and Problem Explanation:**

The given equation is for the general term of an arithmetic sequence: 
\[ a_n = 9 - 5n \]

The problem asks you to find the sum of the first 15 terms of this arithmetic sequence.

### Solution:

1. **Identify the first term** (\( a_1 \)):
   \[
   a_1 = 9 - 5(1) = 9 - 5 = 4
   \]

2. **Identify the second term** (\( a_2 \)):
   \[
   a_2 = 9 - 5(2) = 9 - 10 = -1
   \]

3. **Identify the common difference**:
   The common difference \( d \) is:
   \[
   d = a_2 - a_1 = -1 - 4 = -5
   \]

4. **Sum of the first 15 terms**:
   The formula for the sum of the first \( n \) terms of an arithmetic sequence is:
   \[
   S_n = \frac{n}{2} \times (2a_1 + (n-1)d)
   \]
   Substituting \( n = 15 \), \( a_1 = 4 \), and \( d = -5 \):
   \[
   S_{15} = \frac{15}{2} \times \left( 2(4) + (15-1)(-5) \right)
   \]
   \[
   S_{15} = \frac{15}{2} \times \left( 8 + 14(-5) \right)
   \]
   \[
   S_{15} = \frac{15}{2} \times (8 - 70)
   \]
   \[
   S_{15} = \frac{15}{2} \times (-62)
   \]
   \[
   S_{15} = 15 \times (-31) = -465
   \]

### Final Answer:
The sum of the first 15 terms of the arithmetic sequence is **-465**.

Would you like more details on any step or have other questions? Here are five related questions to explore further:

1. What is the general formula for an arithmetic sequence?
2. How does the common difference affect the behavior of a sequence?
3. How would the sum change if the common difference were positive?
4. Can you derive a formula for the \( n \)-th term of a different arithmetic sequence?
5. How would you find the sum of terms in a geometric sequence?

**Tip:** When working with sequences, always verify the common difference or ratio, as it significantly influences the sequence's behavior.

એક સમાન્તર શ્રેણીનું પદ an = 9 - 5n છે. આ શ્રેણીના પ્રથમ 15 પદોના સરવાળો શોધો.

Learn how to find the sum of the first 15 terms of an arithmetic sequence where the general term is given by an = 9 - 5n. The problem involves understanding arithmetic sequences, finding the common difference, and applying the formula for summation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation