Math 103, Spring 2011, Solutions to Chapter 10B Homework Assignment
Problem 38:
(a)
Since the 7.75% APR is compounded daily, the bank multiplies the account balance by 1 + .0775/365 each
day, which is approximately equal to 1.00021.
Over the course of two years this happens 730 times, since there
are 730 days in two years.
With an initial investment of $874.83, the future value of the account in two years is approximately
($874.83)(1.00021)
730
=
$1021.49
.
(b)
Over the course of one year, the bank multiplies by the daily interest rate 365 times, i.e. by
(1 + .0775/365)
365
, which is approximately equal to 1.080573.
Thus the APY
is
8.0573%
.
Comment:
The APY can be thought of as the effective interest rate per year.
If instead of advertising an APR of
7.75%, and compounding daily, the bank simply offered 8.0573% compounded annually, the impact on your
bank account at the end of the year would be the same.
Thus the APY processes the information in the APR and
the compounding frequency, and summarizes their combined impact by producing a single number, in this case
8.0573%.
Problem 60:
Note that each term in the sum equals the same number 1.075 times the previous term.
This
is why the sum is
geometric, with common ratio 1.075.
Since there are 60 terms in the sum,
N
=60 in the geometric sum formula.
Since the first term in the sum equals $500(1.075), this is the value of
P
in the formula.
Putting this together, the geometric sum formula tells us that the sum equals
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 Spring '07
 Berkowitz
 Math, Addition, $15,000, $120, geometric sum formula

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