Solve the Sequence Formula n²-n: Finding the Fifth Element

Sequence Formulas with Substitution Method

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

n2−n n^2-n

What is the 5th element?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find the 5th element
00:03 Position of the desired element according to the data
00:11 Substitute appropriate values according to the data and solve to find the element
00:18 Always solve exponents first
00:27 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

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

n2−n n^2-n

What is the 5th element?

2

Step-by-step solution

To solve this problem, we will follow these steps:

  • Step 1: Understand the formula given for the sequence.
  • Step 2: Substitute n=5 n = 5 into the formula to find the 5th element.
  • Step 3: Simplify the expression to find the solution.

Now, let's work through each step:

Step 1: We have the formula for the sequence as an=n2−n a_n = n^2 - n .

Step 2: We will substitute n=5 n = 5 into the formula:

a5=52−5 a_5 = 5^2 - 5

Step 3: Calculate the expression:

a5=25−5 a_5 = 25 - 5

a5=20 a_5 = 20

Therefore, the 5th element of the sequence is 20 20 .

3

Final Answer

20 20

Key Points to Remember

Essential concepts to master this topic
  • Formula: Use an=n2−n a_n = n^2 - n where n is the position
  • Technique: Substitute n = 5: 52−5=25−5=20 5^2 - 5 = 25 - 5 = 20
  • Check: Verify by testing: position 5 gives 25−5=20 25 - 5 = 20 ✓

Common Mistakes

Avoid these frequent errors
  • Confusing the position number with the sequence value
    Don't think the 5th element equals 5 = wrong understanding! The position (n = 5) is what you substitute into the formula, not the answer. Always substitute the position number into the given formula n2−n n^2 - n to find the actual sequence value.

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 n2−n n^2 - n actually mean?

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This is the rule that tells you how to find any element in the sequence! The variable n represents the position number, so you substitute the position you want into this formula.

How do I find other elements like the 3rd or 7th?

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Use the same method! For the 3rd element: 32−3=9−3=6 3^2 - 3 = 9 - 3 = 6 . For the 7th element: 72−7=49−7=42 7^2 - 7 = 49 - 7 = 42 . Just substitute the position number for n!

Why do we calculate 52 5^2 first before subtracting?

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We follow the order of operations! Exponents come before subtraction, so 52=25 5^2 = 25 first, then 25−5=20 25 - 5 = 20 .

Can sequence formulas be more complicated than this?

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Yes! You might see formulas like 2n+3 2n + 3 , n3−2n n^3 - 2n , or nn+1 \frac{n}{n+1} . The method is always the same: substitute the position number and calculate!

What if I get a negative number as an answer?

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That's totally fine! Some sequences have negative values. For example, if you tried n=0 n = 0 in this formula: 02−0=0 0^2 - 0 = 0 , which is neither positive nor negative.

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