In the "Carry Out Procedure" section, you are given a sequence, and the instructions are as follows:

1. Given the arithmetic sequence $a_n = 2 + 5n$:
   • a) Find $a_1$: This is the first term in the sequence, which you can calculate by plugging $n = 1$ into the formula.
   • b) Identify the common difference, $d$: In an arithmetic sequence, the common difference is the constant difference between consecutive terms. This is often represented by the coefficient of $n$ in the general formula.

Let's solve these steps:

Step-by-Step Solution

Part (a): Find $a_1$

The formula for the arithmetic sequence is: $a_n = 2 + 5n$ To find $a_1$, substitute $n = 1$: $a_1 = 2 + 5(1) = 2 + 5 = 7$ So, $a_1 = 7$.

Part (b): Identify the common difference $d$

In the given sequence formula $a_n = 2 + 5n$, the term $5$ represents the common difference because it is the value added each time $n$ increases by 1.

Thus, $d = 5$.

Final Answers

• (a) $a_1 = 7$
• (b) The common difference $d = 5$

Would you like more details on this section, or do you need help with other parts of the worksheet?

Additional Practice Questions

1. If $a_n = 2 + 5n$, what is the 10th term, $a_{10}$?
2. For an arithmetic sequence with $a_1 = 3$ and $d = 4$, write the explicit formula for $a_n$.
3. How would you find the sum of the first 15 terms in an arithmetic sequence where $a_1 = 7$ and $d = 5$?
4. If the common difference $d$ were changed to $3$, what would the new sequence formula look like?
5. How can you determine if a sequence is arithmetic by looking at its formula?

Tip:

In an arithmetic sequence, the common difference can be found by subtracting any two consecutive terms.