Arithmetic Sequence: Is 97 in a Series Starting at 18 with Difference 7.5?

Given the series whose first element is 18.

Each term of the series is greater by 7.5 of its predecessor.

Is the number 97 an element in the series?

If so, please indicate your place in the series.

To determine if 97 is an element of the arithmetic sequence, we will use the formula for the $n$-th term:

The $n$-th term of an arithmetic sequence is given by:

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

Where:

• $a_1 = 18$ (the first term)
• $d = 7.5$ (the common difference)
• $a_n = 97$ (the term we are querying)

Substitute these values into the formula to find $n$:

$97 = 18 + (n-1) \cdot 7.5$

Subtract 18 from both sides:

$97 - 18 = (n-1) \cdot 7.5$

$79 = (n-1) \cdot 7.5$

Divide both sides by 7.5:

$\frac{79}{7.5} = n - 1$

Calculate the division:

$n - 1 = 10.5333\ldots$

Add 1 to both sides:

$n = 11.5333\ldots$

Since $n$ must be a whole number (as it represents the position in the sequence), and 11.5333... is not an integer, 97 is not a term in the sequence.

Therefore, the number 97 is not an element in the series.

The correct answer is: No.