Math sequences and series

Sequence

function whose domain is a set of whole numbers

explicitly defined sequence

nth term can be computed only knowing the value of n

recursively defined sequence

nth term can be computed knowing the value of n and the values of one or more previous terms

converging sequence

sequence approaches a unique value

diverging sequence

sequence does not approach a unique value or approaches infinity

series

sum of all the terms in a sequence

what term do sequences start on

n = 0

what term does a series start on

n = 1

when can you use a summation formula

when k = 1

what is the summation formula for: sigma [c]

c * n

summation formula for: sigma [k]

(n(n+1))/2

summation formula for: sigma [k²]

(n(n+1)(2n+1))/6

summation formula for: sigma [k³]

(((n(n+1))/2)²

how do you change the index of a sigma notation

add or subtract to make k = 1, do the same to n, do the opposite to any k’s inside the notation

arithmetic sequence

successive terms have a common difference

arithmetic sequences behave like ________ functions

linear

arithmetic sequence formula when you know the initial term

a[n] = a[0] + dn, a[0] is initial term, d is common difference

arithmetic sequence formula when you have the kth term

a[n] = a[k] + d(n - k), where k is the kth term, d is the common difference

geometric sequence

successive terms have a common ratio

geometric sequences behave like ______ functions

exponential

geometric sequence formula when you have the initial term

g[n] = g[0] * r^n, where g[0] is the initial term and r is the common ratio

geometric sequence formula when you have the kth term

g[n] = g[k] * r^(n-k), where g[k] is the kth term and r is the common ratio

arithmetic series formula

(k) sigma (n=1) [a[1] + d(n-1)]

arithmetic sum formula

S[n] = n/2 * (a[1] + a[n]) - this is the same formula as sigma[k]

Infinite arithmetic series always [diverge/converge] to _______

diverge, negative/positive infinity

geometric series formula

(k) sigma (n = 1) [g[1] * r^(n - 1)]

geometric sum formula

S[n] = g[1] * (1-r^n)/1-r

Formula for compound interest

a[n] = a[n-1] (1 + r/N) + P

a[0] = M
r = interest rate, N = number of deposits made per year, P = additional periodic deposits, M = initial deposit

Formula for loans

a[n] = a[n-1] (1 + r/N) - P

a[0] = B

r = interest rate, N = number of deposits made per year, P = periodic payments, B = initial amount borrowed