# 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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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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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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