Arithmetic Sequence: Is 22.5 the 6th Term After Starting at 10?

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information
• Step 2: Apply the appropriate formula to determine if 22.5 is in the sequence
• Step 3: Check if the computed position is valid

Now, let's work through each step:
Step 1: We know the first term $a_1$ is 10 and the common difference $d$ is 2.5. Our target is 22.5.
Step 2: Using the formula for the $n^{th}$ term of an arithmetic sequence, we have:
$a_n = a_1 + (n-1) \cdot d$ Substituting the known values to check if 22.5 is in the sequence, we set:
$22.5 = 10 + (n-1) \cdot 2.5$ Step 3: Solve for $n$:
$22.5 - 10 = (n-1) \cdot 2.5$ $12.5 = (n-1) \cdot 2.5$ $\frac{12.5}{2.5} = n-1$ $5 = n-1$ $n = 6$ The computation shows $n$ is a positive integer (6), confirming that 22.5 is indeed the 6th term of the series.

Therefore, the solution to the problem is Yes, $6$.