In an arithmetic progression, the common difference is -5. Given the sum of thr first 10 terms of the progression is 45 find the first term of the progression

Let's find the first term of the arithmetic progression given the following information:

- Common difference (\(d\)) = \(-5\)
- Sum of the first 10 terms (\(S_{10}\)) = 45

### Step 1: Use the formula for the sum of the first \(n\) terms

The sum of the first \(n\) terms of an arithmetic progression is given by:
\[
S_n = \frac{n}{2} \times \left(2a_1 + (n-1)d\right)
\]
For \(n = 10\), \(d = -5\), and \(S_{10} = 45\):
\[
S_{10} = \frac{10}{2} \times \left(2a_1 + (10-1)(-5)\right)
\]
\[
45 = 5 \times \left(2a_1 + 9(-5)\right)
\]

### Step 2: Simplify the equation
\[
45 = 5 \times \left(2a_1 - 45\right)
\]
\[
45 = 10a_1 - 225
\]

### Step 3: Solve for \(a_1\)
Add 225 to both sides:
\[
270 = 10a_1
\]
Divide by 10:
\[
a_1 = 27
\]

### Final Answer:
The first term of the arithmetic progression is **27**.

Would you like more details or have any questions?

---

Here are some related questions you might be interested in:

1. How does the negative common difference affect the progression?
2. Can you calculate the sum of the first 20 terms with the same first term and common difference?
3. How do you find the first term if the sum and the last term are given?
4. What is the relationship between the sum, first term, and common difference?
5. How can you determine the number of terms if the sum and common difference are known?
6. What happens to the sum if the common difference is positive instead of negative?
7. How would you find the common difference if only the sum and the first term are given?
8. Can you find the \(n\)th term of this progression?

**Tip:** When dealing with negative common differences, the sequence decreases, which can sometimes result in negative sums if the sequence is long enough.

Learn how to find the first term of an arithmetic progression where the common difference is -5 and the sum of the first 10 terms is 45. This problem involves using the sum formula of an arithmetic progression and solving for the first term. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation