Core Chapter 5: Sequences and Series

Question 1

Define a number sequence

Answer 1

Number sequence: Ordered list of numbers defined by a rule
- Numbers within a sequence: terms

Question 2

Define the general formula (Uₙ = ) in relation to a number sequence

Answer 2

General formula: one way to describe a number sequence, which gives the nth term of the sequence (general term) in terms of n

• U₁: first term of a general sequence
• Uₙ: general (n) term of the sequence

Question 3

Define an arithmetic sequence/arithmetic progression

Answer 3

Arithmetic sequence: sequence where each terms differs from the previous one by the same fixed number (common difference d)
- A sequence is arithmetic if:
Uₙ₊₁ - Uₙ = d

General term formula of an arithmetic sequence:
Uₙ = U₁ + (n-1) d

Question 4

Define a geometric sequence/geometric progression

Answer 4

Geometric sequence: sequence where each term can be obtained from the previous one by multiplying the same non-zero number (common ratio r)
- A sequence is geometric if:
Uₙ₊₁ / Uₙ = r

General formula of a geometric sequence:
Uₙ = (U₁)(rⁿ⁻¹)

Question 5

Explain how geometric sequences can be applied to solve growth and decay questions

Answer 5

Growth and decay: quantity increases/decreases by a fixed percentage of its size each time period
- Uₙ = quantity after n time periods
- U₀ = initial quantity (n=0)
- r = growth/decay multiplier for each time period
- n = number of time periods

Uₙ = (U₀)(r)ⁿ

Question 6

Define a series and state the 2 types of sequences

Answer 6

Series: sum of all the terms in a sequence

Finite sequence (with n terms):
- Sum will always be a finite real number
(Sₙ=U₁+U₂+U₃…+Uₙ)

Infinite sequence:
- Sum cannot be calculated, although the series may converge to a finite number

Question 7

Describe how to use sigma notation

Answer 7

Sigma (Σ): sum
Summation notation: Σ Uₖ
- Lower line: k = __ (lower limit of the summation)
- Upper line: n (upper limit of the summation)
- The sum of all the numbers in the form Uₖ where k = _ , _, _ up to n

Question 8

Define an arithmetic series and find the sum of a finite arithmetic series

Answer 8

Arithmetic series: sum of all the terms of an arithmetic sequence

Sum of a finite arithmetic series:
Sₙ = n/2 (U₁ + Uₙ)
Sₙ = n/2 [2U₁ + (n-1)(d)]

Question 9

Define a geometric series and find the sum of a finite geometric series

Answer 9

Geometric series: sum of all the terms of a geometric sequence

Sum of a finite geometric series:
- When r>1:
Sₙ = [U₁ (rⁿ-1)]/r-1
- When r<1:
Sₙ = [U₁ (1-rⁿ)]/1-r

Question 10

Determine if an infinite geometric sequence is divergent or convergent, and find the limiting sum of a convergent infinite geometric series

Answer 10

• If |r | >1, the infinite sequence is divergent, as rⁿ will become infinitely large
• If |r | < 1 (-1<r<1), the infinite sequence is convergent, as rⁿ will approach 0 as n increases –>(1-rⁿ) will approach 0 –> Sₙ will approach U₁/1-r