Arithmetic Sequence: Is 32.5 the nth Term When First Term is 7 and Difference is 8.5?

To determine if 32.5 is an element in the series, we will follow these steps:

• Step 1: Identify the formula for the $n$-th term of an arithmetic sequence.
• Step 2: Substitute the given values and solve for $n$.
• Step 3: Check if $n$ is a positive integer.

Step 1: We know the $n$-th term formula for an arithmetic sequence is given by:

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

where $a_1 = 7$ and $d = 8.5$.

Step 2: To find if 32.5 is in the sequence, substitute it as $a_n$:

$32.5 = 7 + (n-1) \cdot 8.5$

Simplify and solve for $n$:

$32.5 = 7 + 8.5n - 8.5$

$32.5 = -1.5 + 8.5n$

Add 1.5 to both sides:

$34 = 8.5n$

Divide by 8.5 to solve for $n$:

$n = \frac{34}{8.5}$

$n = 4$

Step 3: Verify that $n$ is a positive integer.

Since $n = 4$ is a positive integer, 32.5 is indeed in the series, and it is the 4th term.

Therefore, the solution to the problem is Yes, $4$.