Calculate the Sixth Element of the Sequence: n² - n

Q: What does 'nth element' or 'nth term' mean in a sequence?

The nth element means the term at position n. So the 6th element means n=6, the 10th element means n=10, and so on. Always use the position number directly in your formula!

Q: How do I know if I calculated the exponent correctly?

For $ 6^2 $, remember that this means 6 × 6 = 36. Don't confuse it with 6 × 2 = 12. The small number is the exponent, telling you how many times to multiply the base by itself.

Q: Why do we subtract n from n² in this sequence?

The formula $ n^2 - n $ is just the rule given for this particular sequence. Different sequences have different rules. Your job is to follow the rule exactly as written, not to question why it's designed that way.

Q: Can I check my answer by calculating other terms?

Absolutely! Try calculating the 5th term: $ 5^2 - 5 = 25 - 5 = 20 $. The sequence should be increasing, so 30 > 20 makes sense for the 6th term.

Q: What if I get a negative number for some terms?

That's perfectly normal! For example, when n=1: $ 1^2 - 1 = 0 $. Some sequences start with zero or negative values and then become positive as n increases.

Quadratic Sequences with Substitution Method

Below is the rule for a sequence in terms of $n$ :

$n^2-n$

What is the value of the 6th element of the series?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's find the 6th term in the sequence.

00:11 We're looking for the position of the term based on the data we have.

00:15 Now, substitute this position into the formula. Then solve to discover the term.

00:22 Always remember to solve the exponent first. It makes things easier!

00:29 And there you have it! That's the solution to our problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Below is the rule for a sequence in terms of $n$ :

$n^2-n$

What is the value of the 6th element of the series?

Step-by-step solution

To solve this problem, we need to find the 6th term of the sequence defined by $n^2 - n$ .

Step 1: Identify the term we need to determine. The problem asks for the 6th element, so $n = 6$ .

Step 2: Use the sequence's formula $a_n = n^2 - n$ .

Step 3: Substitute $n = 6$ into the formula:

$a_6 = 6^2 - 6$

Step 4: Calculate $6^2$ :

$6^2 = 36$

Step 5: Subtract 6 from 36:

$36 - 6 = 30$

Therefore, the value of the 6th element of the sequence is $30$ .

Thus, the correct answer is $30$ .

Final Answer

$30$

Key Points to Remember

Essential concepts to master this topic

Rule: Substitute the term position directly into the given formula
Technique: For n=6: $6^2 - 6 = 36 - 6 = 30$
Check: Verify by calculating a few terms to see the pattern ✓

Common Mistakes

Avoid these frequent errors

Using wrong position number or misunderstanding sequence notation
Don't confuse the 6th element with n=5 or start counting from 0 = wrong answer like 20! The 6th element means n=6 in the formula. Always use the position number exactly as given in the problem.

Practice Quiz

Test your knowledge with interactive questions

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

FAQ

Everything you need to know about this question

What does 'nth element' or 'nth term' mean in a sequence?

The nth element means the term at position n. So the 6th element means n=6, the 10th element means n=10, and so on. Always use the position number directly in your formula!

How do I know if I calculated the exponent correctly?

For $6^2$ , remember that this means 6 × 6 = 36. Don't confuse it with 6 × 2 = 12. The small number is the exponent, telling you how many times to multiply the base by itself.

Why do we subtract n from n² in this sequence?

The formula $n^2 - n$ is just the rule given for this particular sequence. Different sequences have different rules. Your job is to follow the rule exactly as written, not to question why it's designed that way.

Can I check my answer by calculating other terms?

Absolutely! Try calculating the 5th term: $5^2 - 5 = 25 - 5 = 20$ . The sequence should be increasing, so 30 > 20 makes sense for the 6th term.

What if I get a negative number for some terms?

That's perfectly normal! For example, when n=1: $1^2 - 1 = 0$ . Some sequences start with zero or negative values and then become positive as n increases.

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