You have $33,556.25 in a brokerage account, and you plan to deposit an additional $5,000 at the end of every future year until your account totals $220,000. You expect to earn 12% annually on the account. How many years will it take to

11 years

Explanation:

The computation of the number of years that should be taken is shown below:

As we know that

A = P(1 + r%)^n

here

A = future value

P = present value

r = rate of interest

n = time period.

So, the  future value of $33,556.25 is

= $33,556.25× (1.12)^n

In addition to this,  

The  Future value of the annuity is

= Annuity[(1+rate)^time period-1] ÷ rate

= $5,000 [(1.12)^n-1] ÷ 0.12

Now  

$220,000 = $33,556.25 × (1.12)^n + $5,000[(1.12)^n-1] ÷ 0.12

$220,000 = $33,556.25 × (1.12)^n + $41,666.67[(1.12)^n-1]

$220,000 = $33,556.25 × (1.12)^n + $41,666.67 × (1.12)^n - $41,666.67

($220,000 + $41,666.67) = (1.12)^n[$33,556.25 + $41,666.67]

(1.12)^n = ($220,000 + $41,666.67) ÷ ($33,556.25 + $41,666.67)

(1.12)^n = $3.478549866

Now take the log to the both sides

n × log 1.12 = log 3.478549866

n = log 3.478549866 ÷ log 1.12

= 11 years

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