Sequences: Linear & Quadratic

1. What is the $n^{th}$ term?

The $n^{th}$ term is a magic formula that generates the sequence. The letter $n$ stands for the position of the term in the sequence.

2. Linear (Arithmetic) Sequences

A linear sequence goes up or down by the same amount every time. It has a constant first difference.

The "Dino" Method ($dn + c$):
  1. Find the common difference (the gap). This is the number that goes in front of $n$ (let's call it $d$). So you have $dn$.
  2. Work backward to find the "zeroth" term. What number would come before the first term? Add this at the end ($+ c$).
Example 1: Finding the $n^{th}$ term
Find the $n^{th}$ term of the sequence: 5, 8, 11, 14...
Step 1: Find the difference The sequence goes up by 3 each time. So the first part of our rule is $3n$.
Step 2: Find the "zeroth" term To find what comes before 5, subtract 3. $5 - 3 = 2$.
Answer: $3n + 2$

3. Deciding if a term is in a sequence

To check if a number belongs in a sequence, set your $n^{th}$ term expression equal to that number and solve it like an equation to find $n$.

The Integer Rule:
Because $n$ represents a position (1st, 2nd, 3rd...), $n$ must be a positive whole number (an integer). If you solve the equation and get a decimal or fraction, the number is not in the sequence.
Example 2: Is the number in the sequence?
Is 65 a term in the sequence $4n - 3$?
Step 1: Set up the equation $$4n - 3 = 65$$ Step 2: Solve for n Add 3: $4n = 68$
Divide by 4: $n = 17$
Conclusion: Since 17 is a whole number, YES, 65 is the 17th term in the sequence.

4. Quadratic Sequences

A quadratic sequence does not have a constant first difference. Instead, the second difference is constant. The formula always looks like $an^2 + bn + c$.

The $2a, 3a+b, a+b+c$ Method:

Write out the sequence, the first differences, and the second difference. Then use these three mini-equations:

Example 3: Finding a Quadratic $n^{th}$ Term
Find the $n^{th}$ term for: 3, 9, 19, 33...
Step 1: Find the differences Sequence:      3      9      19      33
1st Diff:          6      10      14
2nd Diff:            4        4
Step 2: Use the mini-equations 1) Find a:
$2a = 4 \Rightarrow a = 2$

2) Find b:
$3a + b = 6$
Substitute $a=2$: $3(2) + b = 6$
$6 + b = 6 \Rightarrow b = 0$

3) Find c:
$a + b + c = 3$
Substitute $a=2, b=0$: $2 + 0 + c = 3 \Rightarrow c = 1$
Step 3: Put it all together The format is $an^2 + bn + c$.
$2n^2 + 0n + 1$
Answer: $2n^2 + 1$

10 Practice Questions & Solutions

Try these on paper, showing your steps. Click the question to check your working!

Q1: Find the $n^{th}$ term of the sequence: 7, 11, 15, 19...
1. Difference is $+4$. So it starts with $4n$.
2. The term before 7 would be $7 - 4 = 3$.
Answer: $4n + 3$
Q2: Find the $n^{th}$ term of the sequence: 20, 14, 8, 2...
1. Difference is going DOWN by 6 ($-6$). Starts with $-6n$.
2. Term before 20: $20 - (-6) = 26$.
Answer: $-6n + 26$ (or $26 - 6n$)
Q3: The $n^{th}$ term of a sequence is $5n - 2$. Find the 50th term.
Substitute $n = 50$ into the formula:
$5(50) - 2$
$250 - 2 = 248$
Answer: 248
Q4: Is 88 a term in the sequence $6n + 4$?
Set up the equation: $6n + 4 = 88$
Subtract 4: $6n = 84$
Divide by 6: $n = 14$
Answer: Yes, because 14 is a positive whole number (it is the 14th term).
Q5: Is 45 a term in the sequence $7n - 2$?
Set up the equation: $7n - 2 = 45$
Add 2: $7n = 47$
Divide by 7: $n = 6.71...$
Answer: No, because $n$ is not a whole number.
Q6: Find the $n^{th}$ term for the quadratic sequence: 2, 5, 10, 17...
1st diff: 3, 5, 7
2nd diff: 2

$2a = 2 \Rightarrow a = 1$
$3a + b = 3 \Rightarrow 3(1) + b = 3 \Rightarrow b = 0$
$a + b + c = 2 \Rightarrow 1 + 0 + c = 2 \Rightarrow c = 1$
Answer: $n^2 + 1$
Q7: Find the $n^{th}$ term for the quadratic sequence: 1, 8, 21, 40...
1st diff: 7, 13, 19
2nd diff: 6

$2a = 6 \Rightarrow a = 3$
$3a + b = 7 \Rightarrow 3(3) + b = 7 \Rightarrow 9 + b = 7 \Rightarrow b = -2$
$a + b + c = 1 \Rightarrow 3 + (-2) + c = 1 \Rightarrow 1 + c = 1 \Rightarrow c = 0$
Answer: $3n^2 - 2n$
Q8: Find the $n^{th}$ term for: 4, 15, 32, 55...
1st diff: 11, 17, 23
2nd diff: 6

$2a = 6 \Rightarrow a = 3$
$3a + b = 11 \Rightarrow 9 + b = 11 \Rightarrow b = 2$
$a + b + c = 4 \Rightarrow 3 + 2 + c = 4 \Rightarrow 5 + c = 4 \Rightarrow c = -1$
Answer: $3n^2 + 2n - 1$
Q9: The $n^{th}$ term of a sequence is $n^2 + 4n - 5$. Find the 10th term.
Substitute $n = 10$ into the expression:
$(10)^2 + 4(10) - 5$
$100 + 40 - 5$
$140 - 5 = 135$
Answer: 135
Q10: Challenge! Is 150 a term in the sequence $n^2 + 5$? Prove your answer.
Set up the equation: $n^2 + 5 = 150$
Subtract 5: $n^2 = 145$
Square root both sides: $n = \sqrt{145}$
$\sqrt{145} = 12.041...$
Answer: No. Because 145 is not a perfect square, $n$ is not an integer. Therefore, 150 is not in the sequence.