Question 1

How do I find the next number in a sequence?

Accepted Answer

First, identify the pattern. Check for a constant difference (arithmetic): 3, 7, 11, 15 — differences are all 4, so next is 19. Check for a constant ratio (geometric): 2, 6, 18, 54 — ratios are all 3, so next is 162. Check if it is Fibonacci-like: 1, 1, 2, 3, 5 — each term = sum of two before, so next is 8. Check for square or cubic patterns: 1, 4, 9, 16 — squares, so next is 25. If none of these, look at second differences (quadratic sequences).

Question 2

What is the nth term formula for an arithmetic sequence?

Accepted Answer

For an arithmetic sequence with first term a₁ and common difference d: aₙ = a₁ + (n−1)d. Example: 5, 8, 11, 14, ... a₁ = 5, d = 3. Formula: aₙ = 5 + (n−1)(3) = 5 + 3n − 3 = 3n + 2. Check: a₁ = 3(1)+2=5 ✓, a₂ = 3(2)+2=8 ✓. To find a₁₀₀: 3(100)+2 = 302. This formula lets you find any term without listing all preceding terms.

Question 3

What is the nth term formula for a geometric sequence?

Accepted Answer

For a geometric sequence with first term a₁ and common ratio r: aₙ = a₁ × r^(n−1). Example: 4, 12, 36, 108, ... a₁ = 4, r = 3. Formula: aₙ = 4 × 3^(n−1). a₅ = 4 × 3⁴ = 4 × 81 = 324. To find the sum of n terms: Sₙ = a₁(rⁿ − 1)/(r − 1) for r ≠ 1. For an infinite geometric series with |r| < 1, the sum converges to S∞ = a₁/(1 − r).

Question 4

What is a quadratic sequence?

Accepted Answer

A quadratic sequence has a constant second difference (the differences of the differences are constant). Example: 3, 7, 13, 21, 31. First differences: 4, 6, 8, 10 (not constant). Second differences: 2, 2, 2 (constant!) — this is quadratic. The nth term formula is of the form an² + bn + c. Find a: it equals second difference / 2 = 1. Subtract n² from each term: 2, 3, 4, 5 — this is arithmetic with d=1, giving n+1. So aₙ = n² + n + 1. Check: 1²+1+1=3 ✓.

Question 5

How is the Fibonacci sequence defined?

Accepted Answer

The Fibonacci sequence is defined by the recurrence: F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3. Each term is the sum of the two preceding terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... The ratio of consecutive Fibonacci numbers approaches the golden ratio φ = (1 + √5)/2 ≈ 1.618. Unlike arithmetic or geometric sequences, the Fibonacci sequence does not have a simple linear formula for the nth term, though Binet's formula gives it explicitly: Fₙ = (φⁿ − ψⁿ)/√5 where ψ = (1 − √5)/2.

Type	Pattern	nth Term	Example	Key Test
Arithmetic	Constant difference d	a₁ + (n−1)d	3, 7, 11, 15	Diffs all equal
Geometric	Constant ratio r	a₁ × r^(n−1)	2, 6, 18, 54	Ratios all equal
Quadratic	Constant 2nd diff	an² + bn + c	1, 4, 9, 16, 25	2nd diffs equal
Fibonacci	aₙ = aₙ₋₁ + aₙ₋₂	No simple closed form	1, 1, 2, 3, 5, 8	Each = sum of prev 2
Triangular	Adding 1,2,3,4,...	n(n+1)/2	1, 3, 6, 10, 15	Diffs are 1,2,3,...

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