Core Chapter 5: Sequences and Series Flashcards

1
Q

Define a number sequence

A

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

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2
Q

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

A

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
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3
Q

Define an arithmetic sequence/arithmetic progression

A

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

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4
Q

Define a geometric sequence/geometric progression

A

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)

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5
Q

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

A

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

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6
Q

Define a series and state the 2 types of sequences

A

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

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7
Q

Describe how to use sigma notation

A

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

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8
Q

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

A

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)]

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9
Q

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

A

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

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10
Q

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

A
  • 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
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