P3 - Sequences and series Flashcards
When is a sequence increasing?
each term is greater than the previous
when is a sequence decreasing
each term is less than the previous
When is a sequence periodic
if the terms repeat in a cycle; π’β©π+π=uβ©n for some k.
- k is known as the order
What is an arithmetic sequence
when there is a common difference between each term
What is the form of arithmetic sequences
a, a+ d, a+2d, a+3d, β¦
What is the nth term of an arithmetic sequence given by?
πβ©(π) = π + (π β π)π
What is the last term of an arithmetic sequence given by
(π β π)π
What is an arithmetic series
the sum of terms of an arithmetic sequence
What is the sum of the first n terms of an arithmetic series given by
- πΊβ©π = π/π [ππ + (π β π)π ]
- πΊβ©π = π/π (π + π)
What is a geometric sequence
you must apply a common ratio, r, to get from one term to the next
What is the nth term of a geometric sequence given by
πβπ = ππ^(πβπ)
What is a geometric series
the sum of the terms of a geometric sequence
What is the sum of the first n terms of a geometric series given by?
πΊπ = π(π β π^π)/
π β π
πΊπ = π(π^πβ π)/
π β π
When is the geometric series invalid and why
- when r=1
- because division by zero is undefined
What is the sum to infinity of a geometric sequence?
the sum of the first n terms as n approaches infinity
- this does not exist for all geometric sequences
What is a divergent sequences, with an example?
the sum of a series is not finite, because each term is bigger than the previous
- 2 + 4 + 8 + 16 + 32 + β―
- each term is twice the previous (r=2)
- not finite thus cannot sum to infinity
What is a convergent sequence, with an example
The sum of a such series is infinite, as n gets larger the term will tend to 0
- only convergent if |r|< 1
- 2 + 1 + 1/2 + 1/4 + 1/8 + β―
- each term is half the previous (r=1/2)
What is the sum to infinity of a geometric sequence that only exists for convergent sequences given by?
πΊβ©β = π/
π β π
What is a recurrence relation?
when each term is given as a function of the previous
Using sigma notation, what does the value above the sigma mean?
tells the last value of r for our sequence
Using sigma notation, what does the value below the sigma mean?
- where the series starts
