My Math Forum Need Help with Accumulated Value

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June 20th, 2013, 07:44 PM   #1
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Need Help with Accumulated Value

Hi Guys,

I'm new here.
I have a problem with an exercise equation which I don't understand.
Please have a look at the image:

[attachment=0:27uqbqpq]2013-06-21_101901.gif[/attachment:27uqbqpq]

My question is:

1. I don't think the line 2 is correct, why the 100 are subsctracted and still in the row
2. Where does line 3 coming from??
Attached Images
 2013-06-21_101901.gif (23.7 KB, 898 views)

 June 21st, 2013, 12:34 PM #2 Global Moderator   Joined: May 2007 Posts: 6,558 Thanks: 602 Re: Need Help with Accumulated Value To get from line 1 to line 2, factor out 100(1+i)^4. Line 2 has an error in that all the other 100's shouldn't be there. It looks like a typo. Take them all out and you go from line 2 to line 3 using the standard formula for geometric progression.
 June 21st, 2013, 07:39 PM #3 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 12,906 Thanks: 883 Re: Need Help with Accumulated Value I assume you're solving for i. Why insert the "100"s in there?! Take 'em out! AND, shortening further: let u = 1+i u^4 + u^8 + .... + u^40 = 5(u^4 + u^8 + .... + u^20) : see how unwieldy that is? Using geometric series formula: (u^44 - u^4) / (u^4 - 1) = 5[(u^24 - u^4) / (u^4 - 1) Now doing the math: u^44 - u^4 = 5(u^24 - u^4) u^4(u^40 - 1) = 5[u^4(u^20 - 1)] (u^40 - 1) / (u^20 - 1) = 5 (u^20 - 1)(u^20 + 1) / (u^20 - 1) = 5 u^20 + 1 = 5 u^20 = 4 u = 4^(1/20) = 1.071773.... So i = .071773....
June 21st, 2013, 07:44 PM   #4
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Re: Need Help with Accumulated Value

Quote:
 Originally Posted by Denis I assume you're solving for i. Why insert the "100"s in there?! Take 'em out! AND, shortening further: let u = 1+i u^4 + u^8 + .... + u^40 = 5(u^4 + u^8 + .... + u^20) : see how unwieldy that is? Using geometric series formula: (u^44 - u^4) / (u^4 - 1) = 5[(u^24 - u^4) / (u^4 - 1) Now doing the math: u^44 - u^4 = 5(u^24 - u^4) u^4(u^40 - 1) = 5[u^4(u^20 - 1)] (u^40 - 1) / (u^20 - 1) = 5 (u^20 - 1)(u^20 + 1) / (u^20 - 1) = 5 u^20 + 1 = 5 u^20 = 4 u = 4^(1/20) = 1.071773.... So i = .071773.... ~7.18% cpd annually, I'll bet

June 21st, 2013, 07:57 PM   #5
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Re: Need Help with Accumulated Value

Thanks mathman and dennis for your help.
Actually I'm looking for the X.

Like mathman said, it probably a typo.

Have a look at the complete solution.

[attachment=0:kckc9cf8]2013-06-22_105523.gif[/attachment:kckc9cf8]
Attached Images
 2013-06-22_105523.gif (45.3 KB, 864 views)

June 21st, 2013, 09:42 PM   #6
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Re: Need Help with Accumulated Value

Quote:
 Originally Posted by ajie Have a look at the complete solution.
Your complete solution attachment has: (1 + i)^20 = 4

That's what I gave you in my post: 1 + i = 4^(1/20) : SAME THING!

r = rate every 4 years = (1 + i)^4 - 1 = .3195....

Value end of 40 years = 100(1+r)[(1+r)^10-1] / r = 6194.72

 June 21st, 2013, 10:03 PM #7 Newbie   Joined: Jun 2013 Posts: 3 Thanks: 0 Re: Need Help with Accumulated Value Thanks Denis, I could sleep well now.
 June 22nd, 2013, 06:44 AM #8 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 12,906 Thanks: 883 Re: Need Help with Accumulated Value This can be made really simple, once we have determined that annual rate = 4^(1/20) - 1 = .071773 or 7.1773% This rate is adjusted to coincide with the frequency of the $100 deposits, which is every 4 years: i = [4^(1/20)]^4 - 1 = .319508 or 31.9508% It is now easy to see that the immediate deposit of$100 will accumulate to 131.9508 by the end of first 4-year period. So now we can treat this 131.9508 as a deferred (no immediate deposit) annuity of 131.9508 over 10 periods. So we have: A = 131.9508 n = 10 i = .319508 F = ? (F being Future value) Formula is: F = A[(1 + i)^n - 1] / i so: F = 131.9508[(1 + .31950^10 - 1] / .319508 = 6194.71944... And will look like, in bank statement format: Code: N:YEAR DEPOSIT INTEREST BALANCE 0 : 0 .00 1 : 4 131.95 .00 131.95 2 : 8 131.95 42.16 306.06 3 :12 131.95 97.79 535.80 .... 9 :36 131.95 1,080.62 4,594.72 10:40 131.95 1,468.05 6,194.72

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