Question
Asked Feb 19, 2019
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Sara decides to set up a retirement fund by depositing $24 at the end of each day for 13 years. How much will she have then, if the interest rate is 6.48% compounded weekly and her account starts with $13487 already deposited?

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Expert Answer

Step 1

Let's begin by calculating Effective Interest rate 

Formual for Effective Interest Rate = (1  + (Annual Rate/ number of periods))^ (number of periods)   - 1

                                                                               = (1 + 6.48%/52)^ 52  - 1

                                                                              = 1.066902572   -1   = 0.0669  or 6.69%

Value of $13487 in 13 years will be = $13487 * ( 1 + 6.69%)^13  = $13487 * 2.320715529 = $31299.49

This is what the current sum will grow into. 

 

Step 2

Now, to calculate the next bit one must know the formula for sum of Geometric Progression  given by:

Sum = First term * (1 - common ratio^n)/ (1 - common ratio)

Now $24 collects everyday for a week and then it is compounded. So, every week $168 (24* 7) gets compounded.

First week's collection will get compounded every week for 13 * 52 = 676 weeks at weekly rate

Seconds week collection will get compounded every week for 675 weeks.

So  let's write the Sum of  Geometric progression for this 

Sum = $168 *  ( 1 + 6.48%/52) ^ 676 + $168 *  ( 1 + 6.48%/52) ^ 675 + $168 *  ( 1 + 6.48%/52) ^ 674 + ........................... + $1...

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