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.

Video Solution

Solution Steps

00:00 Is the number 97 in the sequence?

00:04 Let's use the sequence formula

00:14 Let's substitute appropriate values according to the given data and solve

00:21 If the value of N is a positive whole number, then the number is a member of the sequence

00:30 Let's properly open parentheses, multiply by each factor

00:43 Let's group factors

00:54 We want to isolate N

01:30 The value of N is a decimal number, therefore it's not a member of the sequence

01:35 And this is the solution to the question

Step-by-Step Solution

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:

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.