If a bank pays interest at a rate of i compounded m times a year, then the amount of money Pk at the end of k time periods (where one time period =1mth of a year) satisfies the recurrence relation Pk=[1+im]Pk−1 with initial condition P0 = the initial amount deposited. Find an explicit formula for Pn.

Pk=[1+(im)]Pk−1 =[1+(ci{m})][1+(im)]Pk−2 =[1+(im)]2Pk−2 =[1+(im)]3Pk−3 ... =[1+(im)]kPk−k ⇒Pk=[1+(im)]kP0 So, Pn=[1+(im)]n P0 1≥0

Solved: "If a bank pays interest at a rate..."

High School
Algebra
Algebra I
Exponents and radicals

texelaare

Answered question

2021-08-15

If a bank pays interest at a rate of i compounded m times a year, then the amount of money

P_{k}

at the end of

k

time periods (where one time period

= \frac{1}{m} t h

of a year) satisfies the recurrence relation

P_{k} = [1 + \frac{i}{m}] P k - 1

with initial condition

P_{0}

= the initial amount deposited. Find an explicit formula for

P_{n} .

Answer & Explanation

gotovub

Skilled2021-08-16Added 98 answers

$P_{k} = [1 + (\frac{i}{m})] P_{k} - 1$
$= [1 + (\frac{c}{i} {m})] [1 + (\frac{i}{m})] P_{k - 2}$
$= {[1 + (\frac{i}{m})]}^{2} P_{k - 2}$
$= {[1 + (\frac{i}{m})]}^{3} P_{k - 3}$
... $= {[1 + (\frac{i}{m})]}^{k} P_{k - k}$
$\Rightarrow P_{k} = {[1 + (\frac{i}{m})]}^{k} P_{0}$

So, $P_{n} = [1 + (\frac{i}{m})]^{n} P_{0} 1 \geq 0$

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