The sum of the first 8 terms of an arithmetic sequence is 160. If the common difference is 5, what is the first term?

To find the first term \( a_1 \) of an arithmetic sequence where the sum of the first 8 terms is 160 and the common difference \( d \) is 5, we can use the formula for the sum of the first \( n \) terms of an arithmetic sequence:

\[
S_n = \frac{n}{2} \times \left( 2a_1 + (n-1) \times d \right)
\]

Given:
- \( S_8 = 160 \)
- \( n = 8 \)
- \( d = 5 \)

Substitute the values into the formula:

\[
160 = \frac{8}{2} \times \left( 2a_1 + (8-1) \times 5 \right)
\]

Simplify the equation:

\[
160 = 4 \times \left( 2a_1 + 35 \right)
\]

Divide both sides by 4:

\[
40 = 2a_1 + 35
\]

Now, solve for \( a_1 \):

\[
2a_1 = 40 - 35
\]

\[
2a_1 = 5
\]

\[
a_1 = \frac{5}{2} = 2.5
\]

So, the first term \( a_1 \) is \( 2.5 \).

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What is the sum of the first 10 terms of the same sequence?
2. How would the first term change if the sum of the first 8 terms was different?
3. What is the eighth term in the sequence?
4. How can you find the sum of the first \( n \) terms if only the first term and common difference are known?
5. If the common difference were negative, how would that affect the first term?

**Tip:** Understanding the sum formula for arithmetic sequences is crucial, as it allows you to find various unknowns, like the first term, common difference, or even the number of terms.

Learn how to find the first term of an arithmetic sequence when the sum of the first 8 terms is 160 and the common difference is 5. This detailed step-by-step solution covers key algebraic concepts and the arithmetic sequence sum formula, suitable for students in Grades 8-10.

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