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

Q: What does it mean when I get a decimal for the position?

A decimal position number means the number you're looking for doesn't exist in the sequence! Since sequence positions must be whole numbers (1st, 2nd, 3rd...), a decimal like 11.533 tells you the number falls between two actual terms.

Q: What if the number I'm looking for is between two terms?

Then it's not in the sequence! Arithmetic sequences only contain specific values at whole number positions. Numbers that fall between terms are not part of the sequence.

Q: Can the common difference be a decimal like 7.5?

Absolutely! Common differences can be any real number - positive, negative, whole numbers, or decimals. The sequence 18, 25.5, 33, 40.5... is perfectly valid.

Arithmetic Sequences with Non-Integer Positions

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.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:13 Is the number ninety-seven in the sequence? Let's find out.

00:17 We'll start by using the sequence formula. You've got this!

00:27 Now, let's substitute the given values and solve step by step.

00:34 If N is a whole positive number, then ninety-seven is in the sequence.

00:43 Open the parentheses and multiply by each factor carefully.

00:56 Let's group the factors together. You're doing great!

01:07 Now, we want to isolate the variable N. Take your time.

01:43 If N is not a whole number, then ninety-seven isn't part of the sequence.

01:48 And that's how we answer this question. Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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.

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:

$a_1 = 18$ (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.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Use $a_n = a_1 + (n-1) \cdot d$ to find position n
Technique: Solve $97 = 18 + (n-1) \cdot 7.5$ gives n = 11.533...
Check: Position n must be a whole number for valid sequence terms ✓

Common Mistakes

Avoid these frequent errors

Accepting decimal position numbers as valid
Don't conclude that n = 11.533... means 97 is in position 11 or 12 = wrong answer! A decimal position means the number doesn't exist in the sequence since positions must be whole numbers. Always check if your calculated n is a positive integer.

Practice Quiz

Test your knowledge with interactive questions

Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

FAQ

Everything you need to know about this question

What does it mean when I get a decimal for the position?

A decimal position number means the number you're looking for doesn't exist in the sequence! Since sequence positions must be whole numbers (1st, 2nd, 3rd...), a decimal like 11.533 tells you the number falls between two actual terms.

How do I check if my arithmetic sequence formula is correct?

Test with a few known terms! For this sequence: 1st term = 18, 2nd term = 18 + 7.5 = 25.5, 3rd term = 18 + 2(7.5) = 33. If your formula gives the same results, it's correct!

What if the number I'm looking for is between two terms?

Then it's not in the sequence! Arithmetic sequences only contain specific values at whole number positions. Numbers that fall between terms are not part of the sequence.

Can the common difference be a decimal like 7.5?

Absolutely! Common differences can be any real number - positive, negative, whole numbers, or decimals. The sequence 18, 25.5, 33, 40.5... is perfectly valid.

How can I double-check my answer when the result is 'No'?

Calculate a few terms around where your decimal position falls: 11th term = 18 + 10(7.5) = 93, 12th term = 18 + 11(7.5) = 100.5. Since 97 is between 93 and 100.5, it confirms 97 is not in the sequence.

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