What is a sequence?
A sequence is a list of numbers in a fixed order. Each number is called a term. We write the terms as t₁, t₂, t₃ … (read "t one, t two"). The small number is the position.
Examples: 2, 4, 6, 8 … (even numbers), 1, 4, 9, 16 … (square numbers), 1, 1, 2, 3, 5, 8 … (each term is the sum of the two before it).
Explicit and recursive rules
There are two ways to write the rule of a sequence.
- Recursive rule: say the first term, then how to get the next term from the one before. For 3, 5, 7, 9 …: t₁ = 3 and tₙ = tₙ₋₁ + 2. ("Recursive" means "going back": you look back at the last term.)
- Explicit rule: a formula that gives any term straight from its position n. For 3, 5, 7, 9 …: tₙ = 2n + 1. So t₁₀₀ = 201 at once.
A recursive rule is easy to understand. An explicit rule is fast for far-away terms. In step 1 of the 3D, the readout shows both rules for the same towers.
Arithmetic progression (AP) and its nth term
An arithmetic progression (AP) is a sequence where we add the same number every time. That number is the common difference d. The first term is a.
To reach term n from term 1, we add d a total of (n − 1) times. So
aₙ = a + (n − 1)d
Example: 3, 5, 7, … has a = 3, d = 2. The 20th term = 3 + 19 × 2 = 41. d can be negative: 10, 7, 4, 1 … has d = −3. See step 2 in the 3D: blue blocks are a, each orange band is one d.
Sum of the first n natural numbers
What is 1 + 2 + 3 + … + n? Write the sum forwards and backwards and add:
S = 1 + 2 + … + n
S = n + (n − 1) + … + 1
2S = (n + 1) + (n + 1) + … (n times) = n(n + 1)
1 + 2 + … + n = n(n + 1)/2
The 3D (step 3) shows the same idea with blocks: a staircase plus its upside-down copy makes an n by (n + 1) rectangle. So 1 + 2 + … + 100 = 100 × 101 ÷ 2 = 5050.
Geometric progression (GP) and its nth term
A geometric progression (GP) is a sequence where we multiply by the same number every time. That number is the common ratio r. Find it by dividing: r = t₂ ÷ t₁.
To reach term n, we multiply by r a total of (n − 1) times. So
aₙ = a × rⁿ⁻¹
Examples: 1, 2, 4, 8 … (r = 2). 5, 15, 45 … (r = 3). 64, 32, 16 … (r = 1/2, it shrinks). A GP with r bigger than 1 grows very fast. Step 4 of the 3D shows doubling towers.
Fractals: patterns inside patterns
A fractal is a shape where each small part looks like the whole shape. Start with one triangle. Join the midpoints of its sides and remove the middle piece. Now you have 3 smaller triangles. Do the same to each of them, again and again. This is called the Sierpinski triangle.
Number of triangles: 1, 3, 9, 27, … a GP with r = 3. At stage n there are 3ⁿ triangles. The side of each triangle halves every stage: 1, 1/2, 1/4 … another GP with r = 1/2. Step 5 of the 3D builds stages 0 to 3.
Tower of Hanoi
There are 3 pegs. Disks of different sizes sit on the first peg, biggest at the bottom. Goal: move all of them to the last peg. Rules: move one disk at a time, and never put a bigger disk on a smaller one.
Why a recursive rule appears: to move n disks, first move the top (n − 1) disks out of the way, then move the biggest disk once, then move the (n − 1) disks back on top. So Hₙ = 2Hₙ₋₁ + 1, with H₁ = 1. This gives 1, 3, 7, 15, 31 …
Each term is one less than a power of 2, so the explicit rule is Hₙ = 2ⁿ − 1. 3 disks: 7 moves. 10 disks: 1023 moves.
Try it: Tower of Hanoi with coins
Take 3 coins of different sizes (₹1, ₹2, ₹5) and draw 3 circles on paper. Stack the coins on the first circle, biggest at the bottom. Move them to the third circle using the rules. Count your moves. Can you do it in 7? Now add a fourth coin and predict the answer before you try (hint: double and add one). Then check with the disks slider in the last step of the 3D.
Key formulas and definitions
- Recursive rule: give t₁ and how tₙ comes from tₙ₋₁
- Explicit rule: a formula for tₙ in terms of n
- AP: aₙ = a + (n − 1)d, d = a₂ − a₁
- 1 + 2 + 3 + … + n = n(n + 1)/2
- GP: aₙ = a × rⁿ⁻¹, r = a₂ ÷ a₁
- Sierpinski triangle: 3ⁿ triangles at stage n
- Tower of Hanoi: Hₙ = 2Hₙ₋₁ + 1 = 2ⁿ − 1 moves
Worked examples
1. Write the first four terms of the sequence with t₁ = 4 and tₙ = tₙ₋₁ + 5.
t₁ = 4. t₂ = 4 + 5 = 9. t₃ = 9 + 5 = 14. t₄ = 14 + 5 = 19. So 4, 9, 14, 19.
2. Find the explicit rule for 4, 9, 14, 19, … and use it to find t₅₀.
It goes up by 5 each time, so tₙ = 5n + something. For n = 1: 5 + ? = 4, so ? = −1. tₙ = 5n − 1. Check t₂ = 9 ✓. t₅₀ = 250 − 1 = 249.
3. Find the 15th term of the AP 7, 11, 15, …
a = 7, d = 11 − 7 = 4. a₁₅ = 7 + (15 − 1) × 4 = 7 + 56 = 63.
4. Find 1 + 2 + 3 + … + 50.
n = 50. Sum = 50 × 51 ÷ 2 = 2550 ÷ 2 = 1275.
5. Find the 6th term of the GP 3, 6, 12, …
a = 3, r = 6 ÷ 3 = 2. a₆ = 3 × 2⁵ = 3 × 32 = 96.
6. In the Sierpinski triangle, how many small triangles are there at stage 5?
The counts are 1, 3, 9, … so stage n has 3ⁿ. Stage 5: 3⁵ = 243 triangles.
7. How many moves are needed for the Tower of Hanoi with 6 disks? Check with the recursive rule.
Explicit: 2⁶ − 1 = 64 − 1 = 63. Recursive: H₁ = 1, H₂ = 3, H₃ = 7, H₄ = 15, H₅ = 31, H₆ = 2 × 31 + 1 = 63 ✓.
Common mistakes
- Using n instead of (n − 1) in aₙ = a + (n − 1)d or aₙ = a × rⁿ⁻¹. The first term already counts as one step.
- Finding d by subtracting the wrong way (a₁ − a₂). Always do later term minus earlier term.
- Calling a sequence an AP just because it grows. Check the differences: 1, 2, 4, 8 has differences 1, 2, 4, so it is a GP, not an AP.
- Thinking the Tower of Hanoi needs 2ⁿ moves. It is 2ⁿ − 1: for 1 disk it is 1 move, not 2.