Geometric and Quadratic Sequences
Beyond simple linear patterns, geometric and quadratic sequences describe how things grow, shrink, and curve in the real world — from compound interest to projectile paths. Mastering them is a key higher-tier skill.
Geometric vs. Quadratic Sequences
Geometric Sequences
Each term is found by multiplying the previous term by a fixed number called the common ratio ($r$).
2 → 6 → 18 → 54 …
(multiply by 3 each time)
where $a$ is the first term and $r$ is the common ratio.
Quadratic Sequences
The first differences change, but the second difference is constant.
Finding the nth Term of a Quadratic Sequence
To find $an^2 + bn + c$, work out $a$, $b$, and $c$ one at a time using the differences.
Step 1: Find $a$
$a$ is always half the second difference.
Step 2: Find $b$
Use the first term of the first differences.
Step 3: Find $c$
Use the first term of the sequence.
Worked Example: Finding the nth Term
Find the nth term of $3, 7, 13, 21, 31, \ldots$
- Find the differences:
Sequence: 3, 7, 13, 21, 31
1st differences: 4, 6, 8, 10
2nd differences: 2, 2, 2 (constant → quadratic ✓) - Find $a$: $a = \frac{2}{2} = 1$.
- Find $b$: $3a + b = 4 \implies 3 + b = 4 \implies b = 1$.
- Find $c$: $a + b + c = 3 \implies 1 + 1 + c = 3 \implies c = 1$.
- Write the formula: $a_n = 1n^2 + 1n + 1$.
Answer: $a_n = n^2 + n + 1$
Check: $n=1$: $1+1+1=3$ ✓ $n=2$: $4+2+1=7$ ✓ $n=3$: $9+3+1=13$ ✓
Tutor Insights
🤔 Common Misunderstandings
- Confusing sequence types: Always check whether differences are constant (arithmetic), ratios are constant (geometric), or second differences are constant (quadratic).
- The $n-1$ power: Forgetting to write $ar^{n-1}$ — using $ar^n$ instead gives every term wrong.
- Algebraic errors when solving the mini-equations to find $b$ and $c$.
📝 Common Exam Mistakes
- Calculation errors when computing differences or the common ratio.
- Not showing working for finding $a$, $b$, and $c$ — examiners need to see each step.
- Forgetting to substitute $n$ after finding the formula when asked for a specific term.
Practice Questions
- For the sequence $2, 10, 50, 250, \ldots$ find the common ratio and the next two terms.
- Find the nth term of the geometric sequence $100, 50, 25, 12.5, \ldots$
- Find the nth term of the quadratic sequence $2, 5, 10, 17, 26, \ldots$
- The nth term of a quadratic sequence is $n^2 + 2n – 3$. Which term has a value of 12?
Show Answers
- Common ratio $= 5$. Next two terms: $250 \times 5 = \mathbf{1{,}250}$, then $1{,}250 \times 5 = \mathbf{6{,}250}$.
- $a = 100$, $r = 0.5$. Answer: $a_n = 100 \times (0.5)^{n-1}$.
- 2nd diff $= 2 \Rightarrow a=1$. First 1st diff $= 3 \Rightarrow 3+b=3 \Rightarrow b=0$. First term $= 2 \Rightarrow 1+0+c=2 \Rightarrow c=1$.
Answer: $a_n = n^2 + 1$. - $n^2+2n-3=12 \Rightarrow n^2+2n-15=0 \Rightarrow (n+5)(n-3)=0$. Since $n$ must be positive, $n=3$.
Answer: the 3rd term.
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