Question

Susan deposited $2000 into an account with a 3.4% annual interest rate, compounded quarterly. Assuming that no withdrawals are made, how long will ittake for the investment to grow to $2158?Do not round any intermediate computations, and round your answer to the nearest hundredth.圖Yes

Answer 1

Compound Interest

It occurs when the interest is reinvested rather than paying it out. When it happens interest in the next period is then earned on the principal sum plus previously accumulated interest.

The formula is:

`{\displaystyle A=P\mleft(1+{(r)/(n)}\mright)^(nt)}`

Where:

A=final amount

P=initial principal balance

r=interest rate

n=number of times interest applied per time period

t=number of time periods elapsed

We are given the following data:

A = $2000, r = 3.4% annual = 0.034, t = unknown, n = 4 because the interest compounds each quarter, P = $2158

Note we are required to find the time t. Solving for t:

`t=(\log((P)/(A)))/(n\log(1+(r)/(n)))`

Substituting:

`t=(\log ((2158)/(2000)))/(4\log (1+(0.034)/(4)))`

Calculating:

`t=(0.0760347)/(4\cdot0.0036756)=2.25`

It will take approximately 2.25 years for the investment to grow to $2158.