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June 20th, 2013, 08:44 PM  #1 
Newbie Joined: Jun 2013 Posts: 3 Thanks: 0  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]20130621_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?? 
June 21st, 2013, 01:34 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,641 Thanks: 625  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, 08:39 PM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,604 Thanks: 954  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, 08:44 PM  #4  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,604 Thanks: 954  Re: Need Help with Accumulated Value Quote:
 
June 21st, 2013, 08:57 PM  #5 
Newbie Joined: Jun 2013 Posts: 3 Thanks: 0  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]20130622_105523.gif[/attachment:kckc9cf8] 
June 21st, 2013, 10:42 PM  #6  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,604 Thanks: 954  Re: Need Help with Accumulated Value Quote:
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)^101] / r = 6194.72  
June 21st, 2013, 11:03 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, 07:44 AM  #8 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,604 Thanks: 954  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 4year 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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