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?

i would like an answer please :(

OfficialBubbleTanks
Feb 26, 2018

#2**+1 **

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 **

Guest Feb 26, 2018

#3**+1 **

Yes i know the answer, i was wondering if there was a sigma equation to show this

OfficialBubbleTanks
Feb 26, 2018

#4

#10**0 **

∑[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

#5**+1 **

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 ?

ElectricPavlov
Feb 26, 2018

edited by
ElectricPavlov
Feb 26, 2018

edited by ElectricPavlov Feb 26, 2018

edited by ElectricPavlov Feb 27, 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.

Guest Feb 26, 2018

edited by
Guest
Feb 26, 2018

#7**+1 **

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

#8**+1 **

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.

Guest Feb 26, 2018

#14**+1 **

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

#9**+1 **

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

Melody
Feb 27, 2018

#18**+1 **

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

OfficialBubbleTanks
Feb 28, 2018