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.