3 - sequences and series

whats an arithmetic sequence

a list of numbers with a common difference

formula for arithmetic sequence

a + (n-1)d

whats an arithmetic series

where you add up the numbers in an arithmetic sequence

formula for sum of series

n/2 [2a + (n-1)d]

what’s an easier way of thinking of this formula

(a+(n-1)d) = the last term

SO

its n/2 (first term + last term)

proof for sum of arithmetic series formula

1. write out S_n in simple terms

a + (a+d) + (a+2d) + … (a+(n-1)d)

2. write that out backwards, then add the two lists as Gauss’ reversing method stipulates

(2a+(n-1)d) + (2a+(n-1)d) + … (2a+(n-1)d)

3. simplify the sum by saying its n multiples of (2a+(n-1)d) (because you have to add it n times)

n(2a+(n-1)d)

4. divide it all by two as Gauss’ reversing method stipulates

n/2 (2a+(n-1)d)

what is a geometric sequence

a list of numbers with a common ratio

what is the formula for the common ratio

r = current term/prev term

formula for a geometric sequence

ar^(n-1)

what is a geometric series

when you add the numbers in a geometric sequence

formula for sum of geometric series

.

proof for sum of geometric series

1. write out S_n in simple terms

a + ar + ar² + …ar ^(n-1)

2. multiply this equation by r

ar + ar² + ar³ + arⁿ

3. subtract the second equation from the first one (this works bc everything but the end terms cancel out)

S_n - rS_n = a - arⁿ 

4. factorise either side

S_n (1-r) = a (1-rⁿ)

5. divide both sides by (1-r)

S_n = [a(1-rⁿ)]/[1-r]