Interest on principal over 5 years

[yourmom98]

A retired hockey star wants to set up a scholarship fund to assist an underprivileged child who would like to go to a post-secondary institution. He wants to ensure that the student will have $6000 per year for five years. How much should he give to the institution, now, to ensure that this can happen, if the institution is able to invest the money at 10%/a compounded annually?

a)Set up a line diagram showing the present value of each of the $6000 payments.
Note: You will need to find the present value of an amount when calculating the present value of each $6000 payment.

b)Explain why the amounts form a geometric series.

c)Calculate the sum of this geometric series using . Sn=( a(r^n-1)) / (r-1)



a) okay so it goes 6000,6000/(1.1)^1 ... 6000/(1.1)^4 (1+i)=1.1

b) it is geometric because the ratio btwn consecutive terms are constant

c) Okay now i don't know how to sub in and then solve it because i don't know A because i am assuming that he is giving 1 payment for the whole 5 years the kid is going to an university. and i don't know Sn because i am SOLVING for Sn so how do i sub for r and how do i solve?

 

[Fermat]

He is unlikely to give a single payment to cover the student's costs for the whole 5 years but rather 5 separate payments, each one at the start of the currrent year of study.

A Geometric series is,

S = {a, ar, ar², ..., ar^(n-1)}

where a is the initial term,
r is the common ratio

You already used r when creating your line diagram which, in fact, is just the set S.

 

[yourmom98]

well if he does that then it would just be five $6000 payments and the question doesent say WHEN he is giving this money to start saving

 

[Fermat]

According to the question, as you posted it, he gives the money now.

I imagine that means just before the start of the 1st year's study

Although 5 separate payments of $6000 are made, the amount the hockey star gives to the institution for investment is Sn.

 

[yourmom98]

yes that is what got from the question as well that means that there are 5 payments of 6000 to the school. the hockey star pay gives only 1 one payment.

so that means what i did for a) was correct right? my problem is i don't know how to sub that into the equation of Sn

 

[Fermat]

What you did for a) is correct.

The series you got is,

6000,6000/(1.1)^1 ... 6000/(1.1)^4

A Geometric series is,

S = {a, ar, ar², ..., ar^(n-1)}

If you compare your series with the geometric series you will norice that a = 6000 and r = 1/1.1.

Now sub into Sn = ar^(n-1)) / (r-1)