Sequences/Series Flashcards
inductive series
each term is defined relating to its previous term
Arithmetic sequence
A sequence when consecutive terms differ by the addition of a fixed number
a
first term
l
last term
d
common difference
n
number of terms
Geometric sequence
A sequence where each term is found by multiplying the previous term by a fixed number
r
common ratio
geometric sequence example in letters
a, ar, ar^2, ar^3, ar^4…
Periodic sequence
a sequence which repeats itself at regular intervalsp
period
after how many terms does the sequence repeat
increasing
each term in the sequence is greater than the previous term
decreasing
each term in the sequence is less than the previous term
diverging sequence
difference between terms gets greater towards a point
converging
difference between terms gets less towards a point
last term in an arithmetic sequence
l = a +(n-1)d
Summation of an arithmetic sequence with last term
S = 1/2n(a+l)
summation of an arithmetic sequence without last term
S = 1/2n(2a + (n-1)d)
Summation of terms in geometric sequence
S = (a(1-r^n))/(r-1)
OR
S = (a(1-r^n))/(1-r)
Summing geometric sequences to infinity
for -1 0 as n->∞
therefore S = a/(1-r) as n increases
the series converges and has a sum to infinity
examples of geometric growth
compound interest
population growth
radioactive decay
sum of numbers where the first term is 1 and difference is 1 (sum of all natural numbers
1/2n(n+1)
sum of the square of the first n natural numbers
1/6n(n+1)(2n+1)
sum of the cubes of the first n natural numbers
1/4n^2(n+1)^2
Method of differences
- write the series as two or more series
- sub in a few beginning term and a few ending terms
- cancel out the middle terms appropriately
proof by induction for um of series
- prove its true for n=1
- assume true for n=k
- work out n=k+1 by
- show this is equal to k+1 substituted for n
- conclusion
