Patterns & Sequences
Arithmetic adds a constant. Geometric multiplies a constant. Learn to tell them apart first.
A. Arithmetic Sequences
Arithmetic Sequence
A sequence where you add the same number each time. That number is called the common difference (d).
Example3, 7, 11, 15, 19 … → d = 4 (add 4 each time)
Common Difference (d)
d = any term minus the term before it. Can be positive (going up) or negative (going down).
Example20, 15, 10, 5 … → d = 15 − 20 = −5 (going down by 5)
nth Term Formula
Finds any term without listing all of them. a = first term, n = position, d = common difference.
Formulaaₙ = a + (n − 1) × d
Example 1 — Find the 10th term of 5, 9, 13, 17 …
1a = 5, d = 9 − 5 = 4
2a₁₀ = 5 + (10 − 1) × 4 = 5 + 36
a₁₀ = 41
Example 2 — Which term of 3, 7, 11 … equals 79?
1aₙ = a + (n−1)d → 79 = 3 + (n−1) × 4
2(n−1) × 4 = 76 → n−1 = 19 → n = 20
79 is the 20th term
Example 3 — Sum of first 10 terms of 5, 9, 13 …
1Sₙ = n/2 × (2a + (n−1)d)
2S₁₀ = 10/2 × (2×5 + 9×4) = 5 × (10 + 36) = 5 × 46
S₁₀ = 230
B. Geometric Sequences
Geometric Sequence
A sequence where you multiply by the same number each time. That number is called the common ratio (r).
Example2, 6, 18, 54 … → r = 3 (multiply by 3 each time)
Common Ratio (r)
r = any term divided by the term before it.
Example16, 8, 4, 2 … → r = 8 ÷ 16 = ½ (halving each time)
nth Term Formula
a = first term, r = common ratio, n = position.
Formulaaₙ = a × r^(n−1)
Example 1 — Find 5th term of 3, 6, 12, 24 …
1a = 3, r = 6 ÷ 3 = 2
2a₅ = 3 × 2^(5−1) = 3 × 2⁴ = 3 × 16
a₅ = 48
Example 2 — Is 3, 7, 11 arithmetic or geometric?
1Test difference: 7−3=4, 11−7=4 → constant difference
2Test ratio: 7÷3=2.3, 11÷7=1.57 → not constant
Arithmetic (d = 4)
Arithmetic vs Geometric — How They Grow
⚡ MCQ Tip: Arithmetic = constant difference (add/subtract). Geometric = constant ratio (multiply/divide). Always check which type before applying a formula. Most errors happen when students use the arithmetic formula on a geometric sequence.
How to tell them apart:
Try subtracting consecutive terms — if the answer is always the same → arithmetic. Try dividing — if the answer is always the same → geometric.
C. Sum Formulas
Sum of Arithmetic Series
Add n terms of an arithmetic sequence.
FormulaSₙ = n/2 × (2a + (n−1)d)
Or: Sₙ = n/2 × (first + last)
Or: Sₙ = n/2 × (first + last)
Sum of First n Natural Numbers
1 + 2 + 3 + … + n. Most tested special sum.
FormulaS = n(n+1) ÷ 2
Example: 1+2+…+10 = 10×11÷2 = 55
Example: 1+2+…+10 = 10×11÷2 = 55
D. Quick Revision
| Formula | Detail |
|---|---|
AP nth term | a + (n−1)d |
GP nth term | a × r^(n−1) |
AP sum | n/2 × (2a + (n−1)d) |
Sum 1 to n | n(n+1)/2 |
| Arithmetic | Constant DIFFERENCE |
| Geometric | Constant RATIO |