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# Sigma....

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I need to know if there was a better way to do this involving sigma.

To save money for a vacation, you set aside 100 dollars. for each month thereafter, you plan to set aside 10% more than the last month for 12 months. how much money will you have saved up after the 12 months?

i painstakenly calculated all 12 numbers.

100 + 110 + 121 + 133.1 +146.41 + 161.051 + 177.1561 + 194.87171 + 214.358881 + 235.7947691 + 259.37424601 + 285.311670611 = 285.311670611

was there a better and more time saving way to do this?

Feb 26, 2018
edited by OfficialBubbleTanks  Feb 26, 2018

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Feb 26, 2018
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Well, there is a financial formula for that: FV=P{[1 + R]^N - 1/ R},     Where R=Interest rate per period, N=number of periods, P=periodic payment, FV=Future value.

FV = $100 x {[1 + 10%]^12 - 1 / R} FV =$100 x {[1 + 0.10]^12 -1 / 0.10}

FV = $100 x {[1.10]^12 - 1 / 0.10} FV =$100 x {[3.138428376721 - 1 / 0.10}

FV = $100 x {2.138428376721 -1/ 0.10} FV =$100 x       21.38428376........

FV = $2,138.43 Feb 26, 2018 #3 +578 +1 Yes i know the answer, i was wondering if there was a sigma equation to show this OfficialBubbleTanks Feb 26, 2018 #4 +1 Well, you can sum it up like this: ∑[100*1.1^n, n, 0, 11] =$2,138.43

Feb 26, 2018
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∑[100*1.1^n, n, 0, 11] =$2,138.43 Are you saying you can do it like this on a calculator guest? What calculator are you referring to? Melody Feb 27, 2018 #11 +1 Yes Melody. On my personal calculator you can, because it has the "Sigma" notation built into it. That is exactly how I got that answer. Guest Feb 27, 2018 #15 +30907 +1 You can do this on the webcalc calculator as follows: Enter: sum(100*1.1^n,n,0,11) Alan Feb 27, 2018 #16 +110682 0 Thanks Alan :) Melody Feb 28, 2018 #17 +110682 0 Thanks Guest :) Melody Feb 28, 2018 #5 +26620 0 I think this states you start with 100 then for 11 more months add 10% total 12 months correct? The question states: Start with 100 then for 12 more months adds 10 % so would it be n,0,12 ? Feb 26, 2018 edited by ElectricPavlov Feb 26, 2018 edited by ElectricPavlov Feb 26, 2018 edited by ElectricPavlov Feb 27, 2018 #6 +1 Yea, it sums them up like this:$100*1.1^0 + $100*1.1^1 +$100*1.1^2 + $100*1.1^3..............$100*1.1^11 =$2,138.43. Because the first exponent MUST be zero, and the 12th payment would be 11..............., for this:$100*1.1^0 =$100 x 1 =$100, which is the first payment......and so on.

Feb 26, 2018
edited by Guest  Feb 26, 2018
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OK...I think the question is worded incorrectly..... it  STARTS with 100  then for TWELVE MORE months 10% is added......but I think it MEANT 11 MORE months (for a total of 12 months)

ElectricPavlov  Feb 26, 2018
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EP: The reason for the notation{n, 0, 11} is because in financial investments, the first payment is taken to be made at the END of the period, in this case the END of the first month. Example: First pmt. made on Jan.31 of $100. Now, you will wait for the whole of Feb. to get interest on it. So, the END of Feb. you will have:$100 x 1.1 =$110. Then you add another$100 deposit =$210 at the END of Feb. This has exactly the same outcome as the young man calculated in his question:$100, $110,$121....etc.

Feb 26, 2018
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Thanx, I understand that.....BUT, I was merely pointing out that the question is worded improperly.

you start out with 100 and then for 12 MORE months you add 10% (per the question)

'you set aside 100 dollars

'for each month thereafter, you plan to set aside 10% more than the last month for 12 months

So it is a series of 13 payments essentially....first one 100  then twelve more.

It is like saying , in February , I put 100 in the bank....then starting in July I made a deposit of   110

(1st increasing deposit) ...then I did this ELEVEN MORE times for a total of TWELVE more  deposits after the 100.....  Do you see what I am trying to say?    ' for each month thereafter   ....for twelve months'

I see in the poster's answer he only made increasing deposits ELEVEN more times....so what the question should say is you start with 100 then for ELEVEN months you add 10% to the previous month's deposit.

Anyway....minor point.   Just how I read the question.....but apparently no one else did, so I'll go with the flow......

Thanx  .... G'Day !

ElectricPavlov  Feb 27, 2018
edited by ElectricPavlov  Feb 27, 2018
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100 + 110 + 121 + 133.1 +146.41 + 161.051 + 177.1561 + 194.87171 + 214.358881 + 235.7947691 + 259.37424601 + 285.311670611 = 285.311670611

these are the terms of a GP

r= 1.1

this is the best way to present it using sigma notation but it does not give you the answer.

$$\displaystyle \sum_{n=0}^{11} 100*101^n$$

If you want the answer it is easier to do it as the sum of a GP  (that is if you have learned GPs yet)

a=100

n=12     (because n=1 is always the first one)

r=1.1

$$S_{n}=\frac{a(r^n-1)}{r-1}\\ S_{12}=\frac{100(1.1^{12}-1)}{1.1-1}\\ S_{12}=\frac{100(1.1^{12}-1)}{0.1}\\ S_{12}=1000(1.1^{12}-1)\\$$

1000(1.1^12-1) = 2138.428376721

If I take your addition and put it into the web2 calc I get exactly the same answer.

2138.428376721 = 2.138428376721e3 = 2138.428

Feb 27, 2018
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wow i leave a question overnight and get answers galore. thanks guest for giving me the formula

Feb 28, 2018