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Math Help - Geometric series problems - finding future values of annuities

  1. #1
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    Question Geometric series problems - finding future values of annuities

    Hello. I'm having some difficulties with these questions and I'd like to know what I'm doing wrong:

    1. $4300 is deposited at the end of every 6 months for 7 years at 9.5%/a, compounded semiannually. Find the amount of this annuity on the date of the last payment.

    What I'm doing:

    FV = 4300((1.0475)^14) - 1) / .0475
    FV = 4300(0.91494) / .0475
    FV = 3934.266/.0475
    FV = 82,826.66

    But the answer in the back of the book is 81,408.07.

    Also having problems with this question:

    2. At the end of each month, $50 is deposited in an account for 25 years. Then the accumulated money remains in the account for an additional 5 years. Find the amount in the account at the end of this time. The interest rate is 4.8%/a, compounded monthly.

    FV = 50((1.0033^300) - 1)/0.0033
    FV = 50(1.68652)/0.0033
    FV = 84.34258/0.0033
    FV = 25,558.36

    A = 25558.36(1.0033^60)
    A = 25558.36(1.2186)
    A = 31,144.53

    but the answer is $36,724.36. I would really appreciate some help with these questions.
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  2. #2
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    Re: Geometric series problems - finding future values of annuities

    Hi fiftybirds,
    Problem 2
    The sinking fund factor applies to the first 25 years
    Compound amount factor applies to that earned in the next 5 years
    SF S = R (( 1+ i)^n -1 )/i
    CA =(1+i) ^n * S
    n = 300 and i = 0.004 for S
    n = 5 and i =0.048 for CA. I get an answer close to your post
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  3. #3
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    Re: Geometric series problems - finding future values of annuities

    Hello, fiftybirds!

    1. $4300 is deposited at the end of every 6 months
    for 7 years at 9.5%/annum, compounded semiannually.
    Find the amount of this annuity on the date of the last payment.

    What I'm doing:

    \text{FV} \:=\: 4300\,\frac{1.0475^{14} - 1}{0.0475} \:=\:4300\,\frac{0.91494}{0.0475}

    . . . =\: \frac{3934.266}{0.0475} \:=\:82,826.66

    But the answer in the back of the book is 81,408.07.
    I agree with your answer.



    2. At the end of each month, $50 is deposited in an account for 25 years.
    The interest rate is 4.8%/annum, compounded monthly.
    Then the accumulated money remains in the account for an additional 5 years.
    Find the amount in the account at the end of this time.

    . . . . .
    wrong interest rate
    . . . . . . . . . . \downarrow
    \text{FV} \:=\: 50\,\frac{1{\bf.0033}^{300} - 1}{0.0033} \;\hdots

    . . A \:=\: 31,144.53

    But the answer is $36,724.36.

    4.8\%\text{ per year} \;=\;\frac{0.048}{12}\text{ per month} \;=\;0.004\text{ per month}

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    Re: Geometric series problems - finding future values of annuities

    Hello again fiftybirds.
    I rechecked and found mistakes so my new answer is considerably higher.I think my equations are correct
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    Re: Geometric series problems - finding future values of annuities

    Quote Originally Posted by fiftybirds View Post
    FV = 82,826.66
    But the answer in the back of the book is 81,408.07.
    You're correct. Book's answer uses 9%.
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  6. #6
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    Re: Geometric series problems - finding future values of annuities

    Quote Originally Posted by fiftybirds View Post
    A = 31,144.53
    but the answer is $36,724.36.
    Book is correct. You should use 1 + .048/12 = 1.004;
    looks like you did this: 1 + .04/12 = 1.00333....
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  7. #7
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    Re: Geometric series problems - finding future values of annuities

    My use of a calculater this evening is a bit faulty.
    I calculate the sinking fund factor is 578.045, the compound amount factor is 1.26417
    the final amount is 50 x 578.045x 1.26417= $36537.36.If this is wrong I would like to know what I am doing wrong
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    Re: Geometric series problems - finding future values of annuities

    Quote Originally Posted by bjhopper View Post
    ... the compound amount factor is 1.26417
    1.004^60 = 1.27064...
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    Re: Geometric series problems - finding future values of annuities

    Hello again. Thanks for your replies, they really helped me. However I'm now doing present value and I'm having difficulties again... I'm not sure how to approach these problems:

    1. Steven wants to purchase a speedboat that sells for $22,000. The dealer offers a $2000 discount if Steven pays the total amount in cash, or a finance rate of 2.4%/a, compounded monthly, if Steven makes equal monthly payments for 5 years. To pay for the boat with cash now, Steven knows he can borrow the money from the bank at 6%/a, compounded monthly, over the same 5 year period. Which offer should he choose?

    2. Rene buys a computer for $80 down and 18 monthly payments of $55. The first payment is due next month. What is the selling price of the computer system?

    thank you
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    Re: Geometric series problems - finding future values of annuities

    Quote Originally Posted by fiftybirds View Post
    2. Rene buys a computer for $80 down and 18 monthly payments of $55. The first payment is due next month. What is the selling price of the computer system?
    0% rate?

    Note: It is better to start a new thread when posting a new problem.
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  11. #11
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    Re: Geometric series problems - finding future values of annuities

    Quote Originally Posted by fiftybirds View Post
    1. Steven wants to purchase a speedboat that sells for $22,000. The dealer offers a $2000 discount if Steven pays the total amount in cash, or a finance rate of 2.4%/a, compounded monthly, if Steven makes equal monthly payments for 5 years. To pay for the boat with cash now, Steven knows he can borrow the money from the bank at 6%/a, compounded monthly, over the same 5 year period. Which offer should he choose?
    Loan Payment formula: P = Ai / (1 - x) where x = 1 / (1 + i)^n

    P = Payment (?)
    A = Amount borrowed (22000)
    n = number of periods (60)
    i = interest rate per period (.024 / 12)
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