Core Chapter 5: Sequences and Series Flashcards
Define a number sequence
Number sequence: Ordered list of numbers defined by a rule
- Numbers within a sequence: terms
Define the general formula (Uₙ = ) in relation to a number sequence
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
Define an arithmetic sequence/arithmetic progression
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
Define a geometric sequence/geometric progression
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)
Explain how geometric sequences can be applied to solve growth and decay questions
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
Define a series and state the 2 types of sequences
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
Describe how to use sigma notation
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
Define an arithmetic series and find the sum of a finite arithmetic series
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)]
Define a geometric series and find the sum of a finite geometric series
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
Determine if an infinite geometric sequence is divergent or convergent, and find the limiting sum of a convergent infinite geometric series
- 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