Find the 100th Term in the Sequence 10n-9: Linear Pattern Solution

Q: What does the 'n' represent in the formula?

The n represents the position of the term in the sequence. So for the 100th term, n = 100. For the 5th term, n = 5, and so on.

Q: How do I know this is the right formula for the sequence?

Test it! The 1st term: $ 10(1) - 9 = 1 $, 2nd term: $ 10(2) - 9 = 11 $. The difference is always 10, confirming this is an arithmetic sequence.

Q: Why subtract 9 instead of adding?

The formula $ 10n - 9 $ is designed so the sequence starts at 1. If we used $ 10n $, the first term would be 10, not 1. The -9 adjusts the starting point.

Q: Can I use this method for any sequence?

Only if you're given the explicit formula like $ a_n = 10n - 9 $. Some sequences require different approaches, like finding patterns or using recursive formulas.

Q: What if I made an arithmetic error?

Double-check your multiplication: $ 10 \times 100 = 1000 $, then subtraction: $ 1000 - 9 = 991 $. Always verify each step carefully!

Arithmetic Sequences with Formula Substitution

What is the value of the 100th element in the following sequence? $10n-9$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 We are finding the 100th term in the sequence.

00:10 Let's plug in the position of the term into the sequence formula, t hen solve it step by step.

00:18 Substitute N equals 100 into the sequence formula. Remember, N is the term number.

00:33 Always do multiplication and division before addition and subt raction. That's important!

00:45 And there you have it, that's the solution to the question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

What is the value of the 100th element in the following sequence? $10n-9$

Step-by-step solution

To find the 100th element in the sequence defined by $a_n = 10n - 9$ , follow these steps:

Identify that the term we are looking for is the 100th term, so set $n = 100$ .
Substitute $n = 100$ into the formula:
$a_{100} = 10 \times 100 - 9$ .
Perform the arithmetic operations:
$10 \times 100 = 1000$
$1000 - 9 = 991$ .

The value of the 100th element in the sequence is therefore $\textbf{991}$ .

Final Answer

991

Key Points to Remember

Essential concepts to master this topic

Formula: Substitute the term number directly into the given formula
Technique: For $a_{100} = 10(100) - 9 = 991$
Check: Verify the pattern: 1st term = 1, 2nd term = 11, difference = 10 ✓

Common Mistakes

Avoid these frequent errors

Confusing term position with term value
Don't think the 100th term equals 100! The term position (n = 100) goes into the formula, not the answer. Always substitute the position number into the given formula $10n - 9$ .

Practice Quiz

Test your knowledge with interactive questions

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 94,96,98,100,102,104 $$

FAQ

Everything you need to know about this question

What does the 'n' represent in the formula?

The n represents the position of the term in the sequence. So for the 100th term, n = 100. For the 5th term, n = 5, and so on.

How do I know this is the right formula for the sequence?

Test it! The 1st term: $10(1) - 9 = 1$ , 2nd term: $10(2) - 9 = 11$ . The difference is always 10, confirming this is an arithmetic sequence.

Why subtract 9 instead of adding?

The formula $10n - 9$ is designed so the sequence starts at 1. If we used $10n$ , the first term would be 10, not 1. The -9 adjusts the starting point.

Can I use this method for any sequence?

Only if you're given the explicit formula like $a_n = 10n - 9$ . Some sequences require different approaches, like finding patterns or using recursive formulas.

What if I made an arithmetic error?

Double-check your multiplication: $10 \times 100 = 1000$ , then subtraction: $1000 - 9 = 991$ . Always verify each step carefully!

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