Let's work through each part of the problem step by step.

Part (a): Common Difference

The common difference $d$ in an arithmetic sequence can be found by subtracting the first term from the second term.

Given:

• First term ($a_1$) = 3
• Second term ($a_2$) = 7

The common difference $d$ is: $d = a_2 - a_1 = 7 - 3 = 4$

So, the common difference is 4.

Part (b): Eighth Term of the Sequence

The $n$th term of an arithmetic sequence is given by: $a_n = a_1 + (n-1) \times d$

For the eighth term ($a_8$): $a_8 = 3 + (8-1) \times 4 = 3 + 7 \times 4 = 3 + 28 = 31$

So, the eighth term is 31.

Part (c): Sum of the First 15 Terms

The sum $S_n$ of the first $n$ terms of an arithmetic sequence is given by: $S_n = \frac{n}{2} \times (2a_1 + (n-1) \times d)$

For the sum of the first 15 terms ($S_{15}$): $S_{15} = \frac{15}{2} \times (2 \times 3 + (15-1) \times 4) = \frac{15}{2} \times (6 + 56) = \frac{15}{2} \times 62 = 15 \times 31 = 465$

So, the sum of the first 15 terms is 465.

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Do you want any further details or have any questions?

Related Questions:

1. What is the formula for the sum of the first $n$ terms in an arithmetic sequence?
2. How do you find the general term in an arithmetic sequence?
3. What is the difference between an arithmetic and geometric sequence?
4. How would you determine the sum of a sequence if given the first and last terms?
5. How does the common difference affect the growth of an arithmetic sequence?

Tip:

Always double-check the position number $n$ when calculating the $n$th term or the sum of the first $n$ terms in a sequence to avoid simple mistakes.