3. Sequences and Series Flashcards
U
n
The nth term in the sequence
n
The position of the term in the sequence
a
The first term in a sequence
d
The common difference between terms in an arithmetic sequence
U formula (arithmetic) n
a + (n-1)d
S
n
The sum of the first n terms in a sequence
L
The last term in a sequence
S formulae
n
n n
– (2a + (n-1)d) or — (a + L)
2 2
Proving the sum of n terms
- Write as a + (a + d) + (a + 2d) + … + (a + (n-1)d)
- Add the reverse so each element is 2n + (n-1)d
- Multiply by n for 2Sn and divide by 2 for Sn
r
The common ratio between terms in a geometric sequence
U formula (geometric) n
. n-1
a x r
Convergent series
-1 < r < 1
Tend towards a number as you sum
Sum to infinity of a convergent geometric series
. a
S∞ = —–
1 - r
Sn for a geometric sequence
. a(r^n-1) a(1-r^n)
= ———- or ———-
r-1 1-r
Proving the sum of a geometric sequence
Sn = a + ar + ar^2 + ,,, + ar^n-2 + ar^n-1
rSn = ar + ar^2 + ar^3 + … + ar^n-1 + ar^n
Sn - rSn = a - ar^n
Sn (1-r) = a(1-r^n)
Sn = a(1-r^n)/1-r
Sigma notation (a + br)
Find a using r = lower bound
Find d using b
Find n using the amount of values between the upper and lower bound inclusive
Sub into Sn for an arithmetic sequence
Sigma notation (a x b^k-1)
Find a using k = lower bound
Find r with b
Find n using the amount of values between the upper and lower bound inclusive
Sub into Sn for a geometric series
Recurrence relationships
Where a term in a sequence depends on the previous term, you can only find the term after one you know
Recurrence relationships sum of series
You will need to spot a repeated pattern and sum k repeats of that pattern
Periodic sequences
Terms repeat, the order is how many terms are in the pattern that repeats
