① Linear Sequence (Arithmetic)
n-th term: \( a_n = a_1 + (n-1)d \)
Sum of n terms: \( S_n = \dfrac{n}{2}(a_1 + a_n) = \dfrac{n}{2}(2a_1 + (n-1)d) \)
Quick keywords to remember:
CONSTANT DIFFERENCE
d = next − prev
LINEAR = straight line graph
CHECK: a₂−a₁ = a₃−a₂ ?
② Geometric Sequence
n-th term: \( a_n = a_1 \cdot r^{n-1} \)
Sum (r ≠ 1): \( S_n = \dfrac{a_1(1 - r^n)}{1 - r} \) Sum (r=1): \( S_n = na_1 \)
Quick keywords:
CONSTANT RATIO
r = next ÷ prev
MULTIPLY each time
r > 1 → growing
0 < r < 1 → shrinking
③ Other Sequences
Quadratic: 2nd difference is constant
\( a_n = An^2 + Bn + C \)
Fibonacci-type:
\( a_n = a_{n-1} + a_{n-2} \)
The 3-Second Identification Trick:
1st diff. constant → ARITHMETIC
ratio constant → GEOMETRIC
2nd diff. constant → QUADRATIC
add prev two → FIBONACCI
— 1 —
Q 1 ⭐
Find the 10th term of the arithmetic sequence:
\( 3, \ 7, \ 11, \ 15, \ \ldots \)
Identify \(a_1\) and \(d\) first, then plug into \(a_n = a_1 + (n-1)d\)
Q 2 ⭐
The 5th term of an arithmetic sequence is 23 and the common difference is 4.
Find the 1st term and write the formula for the n-th term.
Many students use \(a_5 = a_1 + 5d\) — but it's \(a_1 + (5-1)d\)! Don't forget the "(n−1)"!
Q 3 ⭐⭐
Find the sum of the first 20 terms of the sequence: \(5, \ 9, \ 13, \ 17, \ \ldots\)
📝 Similar Example
Sum of first 10 terms of \(2, 5, 8, \ldots\): \(d=3\), \(S_{10} = \frac{10}{2}(2\cdot2 + 9\cdot3) = 5 \times 31 = 155\)
Q 4 ⭐⭐
The 3rd term of an arithmetic sequence is 11 and the 8th term is 31.
(a) Find the common difference \(d\).
(b) Find the first term \(a_1\).
(c) Which term equals 71?
Set up TWO equations: \(a_3 = a_1 + 2d = 11\) and \(a_8 = a_1 + 7d = 31\). Subtract to eliminate \(a_1\)!
Q 5 ⭐⭐
How many terms are in the arithmetic sequence: \( 7, \ 11, \ 15, \ \ldots, \ 107 \) ?
Set \(a_n = 107\) and solve for \(n\). If \(n\) is not a whole number → something's wrong!
Q 6 ⭐⭐ TRICKY
Insert 4 arithmetic means between 3 and 28.
(i.e. find 4 numbers that, together with 3 and 28, form an arithmetic sequence)
KEY WORD: "k arithmetic means" between a and b
total terms = k + 2
d = (b − a) ÷ (k + 1)
Q 7 ⭐⭐⭐ MOST MISSED
The sum of an arithmetic sequence is \(S_n = 3n^2 + 2n\).
(a) Find \(a_1\), \(a_2\), and the common difference \(d\).
(b) Find \(a_{15}\).
Use \(a_n = S_n - S_{n-1}\) for \(n \geq 2\), but check \(a_1 = S_1\) separately!
📝 Key Formula
\( a_n = S_n - S_{n-1} \) (valid for \( n \geq 2 \))
\( a_1 = S_1 \) always!
— 2 —
Q 8 ⭐
Find the 6th term of: \( 2, \ 6, \ 18, \ 54, \ \ldots \)
Find \(r = \frac{a_2}{a_1}\), then use \(a_n = a_1 \cdot r^{n-1}\)
Q 9 ⭐
Is the sequence \( 4, \ -12, \ 36, \ -108, \ \ldots \) geometric? If yes, state the common ratio \(r\).
Negative ratios are allowed! \(r = \frac{-12}{4} = -3\) ✓ Check: \(\frac{36}{-12} = -3\) ✓
Q 10 ⭐⭐
Find the sum of the first 8 terms of the geometric sequence with \(a_1 = 5\) and \(r = 2\).
\( S_8 = \dfrac{5(2^8 - 1)}{2 - 1} = \ ? \)
Q 11 ⭐⭐ FRACTION TRAP
The 2nd term of a geometric sequence is 12 and the 5th term is \(\dfrac{3}{2}\).
Find the common ratio \(r\) and the first term \(a_1\).
📝 Strategy
Divide term 5 by term 2: \(\dfrac{a_5}{a_2} = r^3\)
So: \(\dfrac{3/2}{12} = r^3\) → \(r^3 = \dfrac{1}{8}\) → \(r = \dfrac{1}{2}\)
Q 12 ⭐⭐
Find the sum to infinity of: \( 16, \ 8, \ 4, \ 2, \ 1, \ \ldots \)
INFINITE SUM
Infinite geometric sum (only when \(|r| < 1\)):
\( S_\infty = \dfrac{a_1}{1 - r} \)
CONDITION: |r| < 1
if |r| ≥ 1 → NO infinite sum!
Q 13 ⭐⭐⭐ MOST MISSED
Three consecutive terms of a geometric sequence are: \( (x-1), \ (x+1), \ (x+5) \)
Find the value of \(x\) and state the three terms.
In a geometric sequence: \(\dfrac{a_2}{a_1} = \dfrac{a_3}{a_2}\), which means \( a_2^2 = a_1 \cdot a_3 \). Cross-multiply!
GEOMETRIC MEAN: b² = ac
always cross-multiply!
Q 14 ⭐⭐⭐ TRICKY
A ball is dropped from a height of 80 m. Each bounce reaches \(\dfrac{3}{4}\) of the previous height.
Find the total distance the ball travels.
📝 Key Insight
Down: \(80 + 60 + 45 + \ldots\) (geometric, \(r = 3/4\))
Up: \(60 + 45 + \ldots\) (same, minus first term)
Total = Down sum + Up sum = \(S_{\infty,\text{down}} + S_{\infty,\text{up}}\)
— 3 —
Q 15 ⭐ Quadratic Sequence
Find the next two terms: \( 1, \ 4, \ 9, \ 16, \ 25, \ \ldots \)
What kind of sequence is this? What is the formula?
These are perfect squares! \(a_n = n^2\)
Next two terms: _______ , _______
Q 16 ⭐ Second Differences
Identify whether the sequence is arithmetic, geometric, or quadratic:
\( 3, \ 6, \ 11, \ 18, \ 27, \ \ldots \)
Show your first and second differences.
📝 How to check
1st differences: \(3, \ 5, \ 7, \ 9, \ldots\) → not constant
2nd differences: \(2, \ 2, \ 2, \ldots\) → constant → Quadratic! ✓
Q 17 ⭐⭐ Quadratic Formula POPULAR
Find the n-th term of the quadratic sequence: \(2, \ 6, \ 12, \ 20, \ 30, \ \ldots\)
Method: \(a_n = An^2 + Bn + C\)
2nd difference \(= 2A\) → find \(A\) first, then use \(n=1,2\) to find \(B\) and \(C\).
Don't forget: \(a_n = n(n+1)\) — always verify with the original terms!
Q 18 ⭐⭐ Mixed — Which Type? IDENTIFICATION
Classify each sequence and find the next term:
(a) \(100, \ 10, \ 1, \ 0.1, \ \ldots\)
(b) \(1, \ 1, \ 2, \ 3, \ 5, \ 8, \ \ldots\)
(c) \(5, \ 11, \ 17, \ 23, \ \ldots\)
(d) \(0, \ 3, \ 8, \ 15, \ 24, \ \ldots\)
GEOMETRIC: ÷ same each time
FIBONACCI: add two prev
ARITHMETIC: + same each time
QUADRATIC: 2nd diff. = const
Q 19 ⭐⭐⭐ Sigma Notation MOST MISSED
Evaluate: \( \displaystyle\sum_{k=1}^{10} (3k - 1) \)
📝 Two Methods
Method 1 (direct): This is arithmetic! \(a_1 = 2\), \(d = 3\), \(n = 10\)
\(S_{10} = \frac{10}{2}(2 + 29) = 5 \times 31 = 155\)
Method 2 (split sigma): \( 3\sum k - \sum 1 = 3\cdot\frac{10\cdot11}{2} - 10 \)
The index starts at \(k = 1\), not \(k = 0\)! Always check the lower limit of the sigma.
Q 20 ⭐⭐⭐ CHALLENGE: Mixed Sums BOSS LEVEL
An arithmetic sequence and a geometric sequence both start with the same first term \(a_1 = 2\).
The arithmetic sequence has \(d = 3\) and the geometric sequence has \(r = 2\).
(a) Write the first 5 terms of each.
(b) Find the first term where the geometric sequence exceeds the arithmetic sequence.
(c) Find \(S_5\) for the arithmetic sequence and \(S_5\) for the geometric sequence.
(d) By how much does the geometric sum exceed the arithmetic sum?
Final Boss Memory Pack:
ARITHMETIC grows slowly (linear)
GEOMETRIC grows FAST (exponential)
geometric always wins eventually!
— 4 —
ARITHMETIC
+ or − same d
\(a_n = a_1+(n-1)d\)
\(S_n = \frac{n}{2}(a_1+a_n)\)
linear graph
GEOMETRIC
× same r
\(a_n = a_1 \cdot r^{n-1}\)
\(S_\infty = \frac{a_1}{1-r}\)
exponential curve
TOP 5 MISTAKES STUDENTS MAKE
❌ Writing \(a_1 + nd\) instead of \(a_1 + (n-1)d\)
❌ Using \(S_\infty\) when \(|r| \geq 1\) (it doesn't exist!)
❌ Forgetting to check \(a_1 = S_1\) separately in sum-to-term problems
❌ Mixing up arithmetic mean (average) vs geometric mean (\(\sqrt{ac}\))
❌ Sigma starting at \(k=0\) vs \(k=1\) — always check!
✦ You've got this. Sequences are all about patterns. Find the pattern, win the problem. ✦
— 5 —