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

Given the series whose first element is 7.

Each term of the series is greater by 8.5 of its predecessor.

Is the number 32.5 an element in the series?

If so, please indicate your place in the series.

Video Solution

Solution Steps

00:00 Is the number 32.5 in the sequence?

00:06 Let's use the sequence formula

00:17 Let's substitute appropriate values according to the given data and solve

00:23 If the value of N turns out to be a positive whole number, then the number is a term in the sequence

00:45 Let's properly expand the brackets and multiply by each factor

01:04 We want to isolate N

01:27 This is the value of N and the position of the term in the sequence

01:34 And this is the solution to the question

Step-by-Step Solution

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

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, $4$ .