The question is asking to determine values for an arithmetic sequence given the first few terms: $19, 15, 11, 7, \dots$.

Step 1: Find the Common Difference ($d$)

In an arithmetic sequence, the common difference is the difference between consecutive terms: $d = a_2 - a_1 = 15 - 19 = -4$

Step 2: Find the Fifth Term ($a_5$)

The formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$ Substitute $n = 5$, $a_1 = 19$, and $d = -4$: $a_5 = 19 + (5-1)(-4) = 19 + 4(-4) = 19 - 16 = 3$

Step 3: General Formula for $a_n$

The general formula is: $a_n = a_1 + (n-1)d$ Substitute $a_1 = 19$ and $d = -4$: $a_n = 19 + (n-1)(-4) = 19 - 4(n-1)$ Simplify: $a_n = 19 - 4n + 4 = 23 - 4n$

Step 4: Find the 100th Term ($a_{100}$)

Use the formula $a_n = 23 - 4n$ with $n = 100$: $a_{100} = 23 - 4(100) = 23 - 400 = -377$

Final Answers:

• $d = -4$
• $a_5 = 3$
• $a_n = 23 - 4n$
• $a_{100} = -377$

Would you like a more detailed explanation of any step?

Related Questions:

1. What is the sum of the first 10 terms of this arithmetic sequence?
2. How do you determine if a sequence is arithmetic or geometric?
3. How can you find the term number of a specific value in an arithmetic sequence?
4. What are real-world applications of arithmetic sequences?
5. How do you derive the general formula for any arithmetic sequence?

Tip: Always verify the common difference to confirm the sequence is arithmetic!