The problem presented in the image asks to "Write a formula for the sequence below, then find a₆."

The sequence given is:

$5, 3, 1, -1, \dots$

Step 1: Identify the pattern

The difference between consecutive terms:

• $5 - 3 = 2$
• $3 - 1 = 2$
• $1 - (-1) = 2$

So, this is an arithmetic sequence with a common difference $d = -2$.

Step 2: General formula for an arithmetic sequence

The general formula for an arithmetic sequence is:

$a_n = a_1 + (n - 1) \cdot d$

Where:

• $a_1 = 5$ (the first term)
• $d = -2$ (the common difference)

Step 3: Find the formula

Substitute $a_1$ and $d$ into the formula:

$a_n = 5 + (n - 1) \cdot (-2)$

Simplify:

$a_n = 5 - 2(n - 1)$ $a_n = 5 - 2n + 2$ $a_n = 7 - 2n$

Step 4: Find $a_6$

To find $a_6$, substitute $n = 6$ into the formula:

$a_6 = 7 - 2(6) = 7 - 12 = -5$

Thus, $a_6 = -5$.

Would you like further explanation or have any questions?

Related Questions:

1. What is the formula for the nth term of a geometric sequence?
2. How can you find the sum of the first n terms of an arithmetic sequence?
3. How do you derive the common difference of an arithmetic sequence from two terms?
4. What happens if the common difference of a sequence is 0?
5. Can arithmetic sequences have fractional or decimal common differences?

Tip: Always check the first few terms of a sequence to identify if it's arithmetic or geometric before finding a general formula!