Answer
Find out the ending balance after one year algebraically.
To proof
Formula of compound interest quarterly
[tex]Compound\ interest_{quarterly} = P( 1+\frac{\frac{R}{4}}{100} )^{4n}[/tex]
where
P = principle
R = rate of interest
n = time period in the years
As given
Mr. Mady opens a savings account with principal P dollars that pays 4.11%
interest compounded quarterly
Time = 1 year
put in the equation
we get
[tex]Compound\ interest\ quarterly = P (1 +\frac{0.0411}{4})^{4}[/tex]
solving the above
[tex]= P ( 1 + 0.010275)^{4}\\= P(1.010275)^{4}[/tex]
Solving
= 1.0417P ( approx )
ending balance after one year is $1.0417P ( approx )
Hence proved
At the end of one year, and at the current rate of increase, Mr. Mady's account would be P x 1.010275⁴
Because the amount is compounded quarterly, the period and the interest rate need to be changed to quarterly figures:
Period = 1 year x 4 quarters = 4 quarters
Interest = 4.11% / 4 quarters = 1.0275% .
The amount after a year is:
= Amount x ( 1 + rate) ^ number of periods
= P x ( 1 + 1.0275%)⁴
= P x 1.010275⁴
In conclusion, the balance after a year is P x 1.010275⁴.
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