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.