Y2, C3 - Sequences & Series Flashcards
What is an arithmetic sequences
A sequence which has a common difference between the terms
What is a geometric sequence
A sequence which has a common ratio between the terms
What is a Fibonacci sequence
A sequence where each term is the previous two terms added together
What does un denote in sequences
the nth term
u3 would be the 3rd term
What does n denote in sequences
The position of the term in the sequence
What does a denote in sequences
The first term
What does d denote in sequences
The difference between terms
What is the formula for the nth tern of an arithmetic sequence
un = a + (n - 1)d
For which values of x would the expression -8, x^2, 17x form the first 3 terms of an arithmetic sequence
Find d (difference)
x^2 - -8 = 17x - x^2
2x^2 - 17x + 8 =0
x = 8, 0.5
What is the formula for the sum of n terms of an arithmetic sequence (formula booklet)
Sn = (n / 2)(2a + (n - 1)d)
What is the alternative formula for the sum of n terms in an arithmetic sequence using the last term (L) (formula booklet)
Sn = (n/2) (a + L)
Find the minimum number of terms for the sum 4 + 9 + 14 + … to exceed 2000
Use formula Sn = (n/2)(2a + (n-1)d)
d = 5, a = 4
4000 < n (8 + 5n - 5)
n = 27.99 therefore n = 28
What is an alternating sequence
A sequence which oscillates between positive and negative
What is the formula for the nth term of a geometric sequence
un = ar^n-1
What does r denote in geometric sequences
The common ratio
The numbers 3, x, x+6 form the first 3 terms of a +ve geometric sequence, find x
Use formula ar^n-1
(x / 3) must equal ((x + 6) / x )
Therefore x = -3 or 6
As sequence +ve, x = 6
What is the formula for the first n terms of a geometric sequence
Sn = (a(1 - r^n)) / (1 - r)
When will an infinite series converge
When lrl < 1
If lrl < 1, what happens as r^n tends to infinity
r^n –> 0
What is the formula for the sum to infinity of a convergent geometric series
S(infinity) = a / (1 - r)
What is a recurrence relation
A term-to-term sequence
The formula tells you the next term in relation to another term e.g.
u(n+1) = 2u(n) + 4
What does it mean if a recurrence relation sequence has a periodic order of 3
The terms in the sequence repeat every 3
How to solve most sigma equations
Sub in first few / last few values and look for patterns
What is a strictly increasing sequence
A sequence where the terms are always increasing
What is a strictly decreasing sequence
A sequence where the terms are always decreasing
What is a periodic sequence
A sequence where the terms repeat in a cycle of order k
What is order k in a periodic sequence
How often the sequence repeats
