Calculate the First Term of the Sequence Using 3n²-4

Q: What if I get a negative number as my answer?

Negative terms are completely normal in sequences! The formula $ 3n^2 - 4 $ gives -1 for the first term, which is correct. Don't worry about the sign - just follow the math.

Q: How can I check if -1 is really the first term?

Calculate a few more terms! Second term: $ 3(2)^2 - 4 = 8 $, Third term: $ 3(3)^2 - 4 = 23 $. The sequence -1, 8, 23... makes sense with increasing values.

Q: Do all quadratic sequences start with negative terms?

No! It depends on the constant term. Here, we subtract 4, so small values of n give negative results. If the formula was $ 3n^2 + 1 $, all terms would be positive.

Q: What's the difference between 3n² - 4 and other sequence formulas?

The $ n^2 $ term makes this a quadratic sequence - terms increase much faster than linear sequences. The pattern of differences between consecutive terms isn't constant, but the second differences are!

Sequence Terms with Quadratic Formulas

A sequence has the rule $3n^2-4$ .

What is the first term?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find member 1

00:03 We'll substitute the appropriate member's position in the formula and solve

00:16 Always solve exponents first

00:23 Always solve multiplication and division before addition and subtraction

00:27 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A sequence has the rule $3n^2-4$ .

What is the first term?

Step-by-step solution

To solve this problem, we'll determine the first term of the sequence using the given formula $3n^2 - 4$ .

Step 1: Identify that the first term corresponds to $n = 1$ .
Step 2: Substitute $n = 1$ into the formula $3n^2 - 4$ .
Step 3: Calculate the value.

Now, let's work through these steps:
Step 1: For $n = 1$ , determine the first term using the sequence formula.
Step 2: Substitute $n = 1$ into $3n^2 - 4$ :

$3(1)^2 - 4 = 3 \times 1 - 4 = 3 - 4 = -1$

Therefore, the first term of the sequence is $-1$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: First term means substitute n = 1 into formula
Technique: Calculate $3(1)^2 - 4 = 3 - 4 = -1$
Check: Verify by computing second term: $3(2)^2 - 4 = 8$ ✓

Common Mistakes

Avoid these frequent errors

Starting with n = 0 instead of n = 1
Don't substitute n = 0 to find the first term = gives 3(0)² - 4 = -4! This assumes sequences start at position zero, but the first term is at position 1. Always use n = 1 for the first term of a sequence.

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

Why does the first term use n = 1 and not n = 0?

In most sequences, we count positions starting from 1: first term, second term, third term, etc. Using n = 0 would give you the 'zeroth' term, which doesn't exist in standard sequences!

What if I get a negative number as my answer?

Negative terms are completely normal in sequences! The formula $3n^2 - 4$ gives -1 for the first term, which is correct. Don't worry about the sign - just follow the math.

How can I check if -1 is really the first term?

Calculate a few more terms! Second term: $3(2)^2 - 4 = 8$ , Third term: $3(3)^2 - 4 = 23$ . The sequence -1, 8, 23... makes sense with increasing values.

Do all quadratic sequences start with negative terms?

No! It depends on the constant term. Here, we subtract 4, so small values of n give negative results. If the formula was $3n^2 + 1$ , all terms would be positive.

What's the difference between 3n² - 4 and other sequence formulas?

The $n^2$ term makes this a quadratic sequence - terms increase much faster than linear sequences. The pattern of differences between consecutive terms isn't constant, but the second differences are!

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