Q: When does a geometric series converge?

When |r| < 1 (the common ratio is between −1 and 1). The sum to infinity is S∞ = a/(1 − r).

Q: What is the sum to infinity of a geometric series?

S∞ = a/(1 − r), valid only when |r| < 1. As n → ∞, the terms become negligibly small.

Q: What is sigma (Σ) notation?

Shorthand for a sum: Σ from r=1 to n of f(r) means add up f(1) + f(2) + ... + f(n). The variable r is the index of summation.

Q: What is the Fibonacci sequence?

A sequence where each term is the sum of the two preceding terms: 1, 1, 2, 3, 5, 8, 13, 21... The ratio of consecutive terms approaches the golden ratio (φ ≈ 1.618).

Q: What is the binomial expansion?

(1 + x)ⁿ = 1 + nx + n(n−1)/2! × x² + n(n−1)(n−2)/3! × x³ + ... Valid for all n when |x| < 1, or for positive integer n for all x.

Q: What is the binomial coefficient ⁿCᵣ?

ⁿCᵣ = n! / (r!(n−r)!) — the number of ways of choosing r items from n. Used in the binomial expansion: (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ.

Question 1

What is an arithmetic sequence?

Accepted Answer

A sequence with a constant difference (d) between consecutive terms. General term: aₙ = a + (n−1)d, where a is the first term.

Question 2

What is the formula for the sum of an arithmetic series?

Accepted Answer

Sₙ = n/2 × (2a + (n−1)d) or Sₙ = n/2 × (a + l), where l is the last term.

Question 3

What is a geometric sequence?

Accepted Answer

A sequence where each term is multiplied by a constant ratio (r). General term: aₙ = arⁿ⁻¹.

Question 4

What is the formula for the sum of a geometric series?

Accepted Answer

Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1, or Sₙ = a(rⁿ − 1)/(r − 1). Use the first form when |r| < 1.

Question 5

When does a geometric series converge?

Accepted Answer

When |r| < 1 (the common ratio is between −1 and 1). The sum to infinity is S∞ = a/(1 − r).

Question 6

What is the sum to infinity of a geometric series?

Accepted Answer

S∞ = a/(1 − r), valid only when |r| < 1. As n → ∞, the terms become negligibly small.

Question 7

What is sigma (Σ) notation?

Accepted Answer

Shorthand for a sum: Σ from r=1 to n of f(r) means add up f(1) + f(2) + ... + f(n). The variable r is the index of summation.

Question 8

What is the Fibonacci sequence?

Accepted Answer

A sequence where each term is the sum of the two preceding terms: 1, 1, 2, 3, 5, 8, 13, 21... The ratio of consecutive terms approaches the golden ratio (φ ≈ 1.618).

Question 9

What is the binomial expansion?

Accepted Answer

(1 + x)ⁿ = 1 + nx + n(n−1)/2! × x² + n(n−1)(n−2)/3! × x³ + ... Valid for all n when |x| < 1, or for positive integer n for all x.

Question 10

What is the binomial coefficient ⁿCᵣ?

Accepted Answer

ⁿCᵣ = n! / (r!(n−r)!) — the number of ways of choosing r items from n. Used in the binomial expansion: (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ.

Sequences and Series Flashcards

Free A Level Maths Revision Cards

See all 20 cards for free

Sequences and Series Flashcards

Free A Level Maths Revision Cards

See all 20 cards for free

More free Maths flashcards