Question

An engineer deposits $8000 in year 1, $8500 in year 2, and amounts increasing by $500 per year through year 10. At a compounding interest of 10%, the future worth in year ten is?

Answer 1

I want to say it's 14,300 dollars I'm not certain but if you multiply 500 by 10 you get 5,000 and add that to the existing balance which is 8,000 so it would become 13,000 and 10% interest of that I'm pretty sure is 1,300 and add that to 13,000 would make it 14,300. I'm not sure you should double check but that's what I got.

Answer 2

no, I"m sorry but that's not within the choices. :(

Answer 3

not sure what to do with this either. How did you get your answer by the way?

Answer 4

Is 14,718 one?

Answer 5

Wait, nevermind I don't think it's right, sorry.

Answer 6

i saw that the answer is 60,600 but idk how it happened. lol

Answer 7

oh nevermind that's wrong. different question.

Answer 8

I mean it's not too bad, but I guess you have to use some formula..compound interest? And why is 60.6k wrong?

Answer 9

yeah, i've been using that but i really don't know if it's just as simple as plugging the values in directly.

the question I saw was asking for present worth at 0 year

Answer 10

There's a bit more to it then plug and chug since we have the increase in amount by 500

Answer 11

So, what formula are you using?

Answer 12

still the compound interest. but what i tried was getting the value for each. for example, 8k then 8500, then 9000 etc... then i added those but that was wrong. lol

Answer 13

This correct?

Answer 14

yes, for compound interest.

Answer 15

Ok gotcha, I'm not in business but I'm sure we can figure it out, I guess we can simply do this calculation twice but the second time with 500 being the principle amount and add it to our first that could do the trick.

Answer 16

Lets list of the what the variables are first, what is r and n?

Answer 17

i think r=10=0.10 and n=1 it only said compounded per year, right?

Answer 18

Looks this is an annuity problem

Answer 19

Ooh!

Answer 20

i was also thinking of that, but wasnt sure what to do.

Answer 21

Lets try and get a recurrence relation

Answer 22

are you good with solving recurrence relations ?

Answer 23

no. sorry. :(

i just know the basic formula for annuity

Answer 24

What's the formula for it

Answer 25

I think it makes more sense

Answer 26

well, for future worth it's this :

Answer 27

\[FV = A (\frac{ ((1 + interest)^n - 1) }{ i })\]

Answer 28

FV= future value A=annuity

This is it if im not mistaken

Answer 29

interest rate sorry not interest

Answer 30

that only works if the deposits are fixed amounts

but in your problem, the deposts are increasing by 500 each year

Answer 31

do it twice second time with 500!

Answer 32

then add it

Answer 33

doesn't work, 500 isn't fixed

500, 1000, 1500, ...

Answer 34

i tried that it was wrong hahaha

Answer 35

ooh

Answer 36

Hey reccurance relation is a great idea

Answer 37

Lets derive a new formula, it will be easy, I promise you

Answer 38

Let the amount in bank after \(n\) years be \(B_n\)

Answer 39

Can you tell me what would be the amount in bank after \(n+1\) years ?

Answer 40

Bank adds 10% interest rate, so it should be \(1.1B_n\)

Also, you're depositing an amount of \(8000 + 500n\) right ?

Answer 41

Overall, the amount after \(n+1\) years is given by :
\[B_{n+1} = 1.1B_n + 800 + 500n\]

Answer 42

see if that makes sense...

Answer 43

if that doesn't make sense, excel is always there for us :)

Answer 44

Fixed the typo :

Overall, the amount after \(n+1\) years is given by :
\[B_{n+1} = 1.1B_n + 800\color{red}{0} + 500n\]

Answer 45

Haha nice

Answer 46

@1018 you're not expected to understand above relation if you're not introduced to recurrence relations yet...

Answer 47

man i'm really sorry i just thought i should know that by now. is recurrence relations part of engineering economics?

Answer 48

continue please. haha. me head hurts but in a good learning kind of way. lol

Answer 49

I don't want this thread to be your introduction to recurrence relations haha
lets use excecl instead

Answer 50

hey can you teach me with this one

https://www.coursehero.com/file/6427211/Test-1-section-011-Version-A-ANSWERS/

Answer 51

look at number 4, it has the same almost question, but it's asking for present worth. i dont' understand the solution though. isn't that the one for excel?

Answer 52

they are solving the problem using @Astrophysics method

Answer 53

work it in two steps :
1) future worth of annual payments of 8000
2) future worth of increasing annual payments of 500

Answer 54

for #1 you can directly use your annuity formula

Answer 55

oooh. how bout for 2?

Answer 56

for number 2 also we may use the same annuity formula, but multiple times

Answer 57

Can you tell me the deposit amount in year 10 ?

Answer 58

it should be 8000 + 500*9, right ?

Answer 59

Think of these 500 deposits like this :

1) 500 annuity for 9 years
2) 500 annuity for 8 years
....
9) 500 annuity for 1 year

Answer 60

Adding all those 9 annuities gives you :

\[\dfrac{500(1+0.1)^9}{0.1} +\dfrac{500(1+0.1)^8}{0.1} +\cdots + \dfrac{500(1+0.1)^1}{0.1} \]

Answer 61

simplify a bit and get below geometric series :

\[5000(1.1^9+1.1^8+\cdots + 1.1^2 + 1.1^1) \]

Answer 62

remember the partial sum formula for geometric series ?

Answer 63

Notice that the sum in parenthesis is a geometric series with \(a = 1.1\), \(r=1.1\), and \(n=9\)

Answer 64

use below formula to work the sum

Answer 65

hey please continue haha, im just looking at you explain. thanks a lot!

Answer 66

i plugged the values in i got 14.937 ?

Answer 67

simplify a bit and get below geometric series :

\[5000(1.1^9+1.1^8+\cdots + 1.1^2 + 1.1^1) \]

that evaluates to
\[5000*1.1*\dfrac{1.1^9 - 1}{1.1-1}\]

simplify

Answer 68

Yes, you forgot the 5000 at the front ;)

Answer 69

it says 74687.123

Answer 70

Looks perfect!
so the future value of those 500 increasing deposits is 74687.123 at the end of 10th year

Answer 71

thanks! but wait i have to add this to the 8000 annuity right?

Answer 72

Is 74687.123 definitely the answer?
I calculate a much higher total.

Answer 73

74687.123 is only the 500 deposits part :

```
work it in two steps :
1) future worth of annual payments of 8000
2) future worth of increasing annual payments of 500
```

still need to work step1 and add to above amount..

Answer 74

yup with the future value of 8000 right? then that's it?

Answer 75

I did it by figuring each year's totals.
Example
8,000 in the first year yields after NINE years
8,000 * (1.1)^9 = 18,863.58

Answer 76

nice, @wolf1728 are you getting 202186.5 for the final answer ?

http://www.wolframalpha.com/input/?i=8000*%28%281%2B0.1%29%5E10-1%29%2F0.1+%2B+5000*1.1*%281.1%5E9-1%29%2F%281.1-1%29

Answer 77

yes @1018

Answer 78

I'm wondering whether the first year's deposit should be calculated for 9 or ten years.
Anyway, if it's for 9 years, then total = 157,186.52
If the 8,000 gets calculated for 10 years, then total = 172,905.17

Answer 79

Interesting, is that only the 8000 part ?

Answer 80

No, I'm thinking it is for all the deposits.
So, 8,500 is calculated for 9 years at 10%
So, 9,000 is calculated for 8 years at 10%
I have the whole list (I use Open Office - the free version of Excel)

Answer 81

Assuming deposits are made at the start of the year, do we have

```
8000 for 10 years
8500 for 9 years
9000 for 8 years
.
.
.
12500 for 1 year
```

?

Answer 82

That's how I did it.
By the way, the formula you used, is it correct?

Answer 83

Here are my yearly totals:
8,000.00 20,749.94
8,500.00 20,042.56
9,000.00 19,292.30
9,500.00 18,512.81
10,000.00 17,715.61
10,500.00 16,910.36
11,000.00 16,105.10
11,500.00 15,306.50
12,000.00 14,520.00
12,500.00 13,750.00


TOTAL 172,905.17

Answer 84

yeah your work looks flawless to me
there must be some mistake in my method above hmm

Answer 85

You'd think a problem this complex would be supplied with an answer.