2. Suppose $13$ and counting by $8$ is possible that the numbers $85$ is in this sequence?

Key takeaway: This is an arithmetic sequence membership check. The key is to write a term rule and test whether $85$ fits it exactly.

Step 1: Write the sequence rule

Starting at $13$ and counting by $8$ gives an arithmetic sequence:

$13, 21, 29, 37, \dots$

The $n$th term is

$a_n = 13 + 8(n-1).$

Step 2: Test whether $85$ is a term

Set the formula equal to $85$:

$13 + 8(n-1) = 85$

$8(n-1) = 72$

$n-1 = 9$

$n = 10$

Step 3: Conclusion

Because $n=10$ is a whole number, $85$ is in the sequence.

You can also check by listing terms:

$13, 21, 29, 37, 45, 53, 61, 69, 77, 85$

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Pitfalls the pros know 👇 A common mistake is dividing $85-13$ by $8$ and forgetting to check whether the result gives a whole-number term position. For arithmetic sequences, the index must come out as a positive integer.

What if the problem changes? If the sequence started from a different first term or used a different common difference, you would still check membership the same way: solve $a_1 + (n-1)d = 85$ and see whether $n$ is a whole number.

Tags: arithmetic sequence, common difference, nth term