P2.3 Sequences & Series Flashcards
what is the difference b/w consecutive terms in arithmetic sequences?
constant = common difference
if common difference is +ve, the sequence is increasing
if common difference is -ve, the sequence is decreasing
what is the formula for nth term of an arithmetic sequence?
Un = a + (n-1)d
a - first term
d - common difference
what is an arithmetic series?
the sum of the terms of an arithmetic sequence
what is the sum of the first n terms of an arithmetic series?
Sn = n/2 (2a + (n-1)d)
Sn = n/2 (a+l)
what is the method to prove the sum of the first n terms of an arithmetic sequence?
- write out the sum of the terms
Sn = a + (a+d) + (a+2d)…(a+(n-2)d) + (a+(n-1)d) - write out the sum reversed
Sn = (a+(n-1)d) + (a+(n-2)d) + (a+2d) + (a+d) + a - add together & divide by 2
2Sn = n(2a + (n-1)d)
Sn = n/2 (2a + (n-1)d)
see pg 63 P2 textbook
what is a geometric sequence?
has a common ratio b/w consecutive terms
to get from one term to the next, multiply by the common ratio
what is the formula for the nth term of a geometric sequence?
Un = ar^(n-1)
a is the first term
r is the common ratio
using logs to find n
what is the formula for the sum of the first n terms of a geometric series?
Sn = a(1-r^n) / 1-r when r<1
Sn = a(r^n-1) / r-1 when r>1
what is the proof for the formula for the sum of the first n terms of a geometric series?
see OneNote/p71 P2 textbook
what happens when r is -ve?
it creates an alternating sequence
what is true for in a convergent sequence?
|r| < 1
terms tend towards a limit
what is true for a divergent sequence?
|r| > 1
S∞ for a divergent sequence is ∞
what is the formula for the sum to infinity for a convergent series?
S∞ = a / 1-r
prove the sum to infinity formula
see OneNote/pg 73 P2 textbook
what does sigma signify
a sum
limits written on the top & bottom of the symbol Σ to show which terms you are summing
describe how to solve problems involving Σ
& practice
sub in values of r to find the terms of the sequence - r does not have to start at one
see OneNote
what is a recurrence relation?
the rule to get from one term to the next term
Un+1 = f(Un) defines each term of a sequence as a function of the previous term
increasing vs decreasing sequences
increasing if Un+1 > Un
decreasing if Un+1 < Un
describe periodic sequences
sequence with terms that repeat in a cycle
there is an integer k such that Un+k = Un
k is the order of the sequence
always write out the first few terms of the sequence
describe modelling with geometric sequences & series
geometric sequence: increasing/decreasing by the same percentage each year
geometric series: models the amount in total over n years
modelling with series practice
