Formulas
i Sum of an arithmetic sequence: If a1 , a2 · · · is an arithmetic sequence where at =
at−1 + d, then
n
a1 + a2 + · · · + an = (a1 + an )
2
ii Sum of a geometric sequence: If a1 , a2 · · · is a geometric sequence where at = dat−1 ,
then
dn − 1
a1 + a2 + · · · + an = a1
d−1
iii Simple interest: S = P (1 + rt).
If you invest P dollars into an account that pays a simple interest rate r, after t years your
account will be worth S dollars.
iv Compound interest: S = P 1 +
r nt
.
n
If you invest P dollars into an account that earns an interest rate r, compounded n times each
year, after t years your account will be worth S dollars.
v AP Y = 1 +
r n
n
− 1 or AP Y = er − 1.
The APY of an account is the simple interest rate that yields the same return.
vi Future value of an ordinary annuity S =
R((1+rc )N −1)
,
rc
where rc = nr , N = nt.
Given an account that pays an interest rate r, compounded n times each year, if you deposit a
fixed amount of money R into it at the end of each of the n compounding periods, after t years
your account will be worth S dollars.
vii Future value of an annuity due: S = R
h
(1+rc )N +1 −1
rc
i
− R, where rc = nr , N = nt.
Given an account that pays an interest rate r, compounded n times each year, if you deposit a
fixed amount of money R into it at the beginning of each of the n compounding periods, after t
years your account will be worth S dollars.
viii Present value of an ordinary annuity: S =
R(1−(1+rc )−N )
,
rc
where rc = nr , N = nt.
Given an account that pays an interest rate r, compounded n times each year, if it initially
contains S dollars, you will be able to withdraw a fixed amount of money R from it at the end
of each of the n compounding periods for t years.
ix Present value of an annuity due: S =
R(1+rc )(1−(1+rc )−N )
,
rc
where rc = nr , N = nt.
Given an account that pays an interest rate r, compounded n times each year, if it initially
contains S dollars, you will be able to withdraw a fixed amount of money R from it at the
beginning of each of the n compounding periods for t years.
x Present value of a deferred ordinary annuity: P =
N = nt.
R(1−(1+rc )−N )
,
rc (1+rc )m
where rc = nr ,
Given an account that pays an interest rate r, compounded n times each year, if it initially
contains S dollars and you don’t withdraw any money during the first m compounding periods,
you will be able to withdraw a fixed amount of money R from it at the end of each of the n
compounding periods for t years.
xi Amortization formula: SN −k = R
1−(1+rc )−(N −k)
rc
, where rc = nr , N = nt.
If you want to pay off a loan by making n payments of R dollars each year for t years, the
unpaid balance after making k payments will be of SN −k dollars.
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