Question

If your starting salary were $50,000 and you received a 4% increase at the end of every year for 15 years, what would be the total amount, in dollars, you would have earned over the first 16 years that you worked?

Answer 1

I know the sequence would be.
a_N = 50000(1.04)^n-1
but it says it ends after 15 years so i'm a bit confused at that part

Answer 2

are you wanting a final equation or just the solution?

Answer 3

well i need the the total amount. so the solution, but knowing how to get to that solution is much appreciated as well.

Answer 4

would the 16th year just be the 50000 added?

Answer 5

ok
so the 4% yearly increase ends at 15 years, and so the salary that would be paid out at the end of the 16th year would be the same salary that was paid out at the end of the 15th year.

Answer 6

are you in calculus?

Answer 7

I'm taking pre-calc atm, but learning calc by myself.

Answer 8

okay so ,
the 15th year is
a_N = 50000(1.04)^n-1
for the 15th year..
n = 16.
a_16 = 50000 (1.04)^15
a_ 16 = 90047 dollars
but then its a sum of all of them.

Answer 9

so then I would have to use the geometric sum formula.

Answer 10

a_N = 50000(1.04)^15-1
=90046.17
is the amount at the end of year 15 that will be paid to him at that year.
since his his increase stops after year 15, then his salary at the end of year 16 will also be 90046.17

to get the total amount of money made for all the years,
take the integral of your function:
= 50000(1.04)^n-1 and evaluate it from n=0 to n=15, then add the value of 90046.17 to make up for the 16th year

Answer 11

\[50000\frac{ 1-(1.04)^{15} }{ 1-1.04 }\] ?

Answer 12

using the formula.
\[a \frac{ 1-r^n }{ 1-r}\]

Answer 13

wait i got the ratio wrong lol.

Answer 14

mmm.. personally, i'm not familiar with that equation, but plug in you numbers and let me know what you get

Answer 15

i think.

Answer 16

oh, and i was wrong, i punched it in wrong, its not 90046, its 86583.82 for the salary of year 15 and 16

Answer 17

In fact isn't this equivalent to the way an annuity works?

Answer 18

I have no idea .-.

Answer 19

so um any ideas. ?..

Answer 20

if not , i'll just email my teacher bout it . and ask for help that way.

Answer 21

50000 + 50000(1+r) + 50000(1+r)^2 + 50000(1+r)^3 + ...... + 50000(1+r)^15 = ?

Answer 22

would the ratio just be 0.04 or 1.04? though?..

Answer 23

I think the ratio is 1.04

Answer 24

50000(1 + 1.04 + 1.04^2 + 1.04^3 + .... + 1.04^15)

Answer 25

I've got it figured out year by year on Excel.
I could post the figures right here.

50,000.00
52,000.00
54,080.00
56,243.20
58,492.93
60,832.65
63,265.95
65,796.59
68,428.45
71,165.59
74,012.21
76,972.70
80,051.61
83,253.68
86,583.82

1,001,179.38

Answer 26

That's for 15 years

Answer 27

Alright thanks, doing this by hand would've taken much longer..
would you then just add 50000 for the 16th year?
or add 86.553.82?

Answer 28

I don't know. Well, you've got 15 years anyway.

Answer 29

50000(1 + 1.04 + 1.04^2 + 1.04^3 + .... + 1.04^15)
= 50000 * (1.04^15 - 1) / (1.04 - 1) = 1,001,179.38

Answer 30

oh, i got the original formula wrong -.-

Answer 31

ranga - great minds think alike LOL

Answer 32

so erm, who do i medal

Answer 33

medal ranga for his hard work if you could wolf lol

Answer 34

to wolf. he got it first.

Answer 35

medal whom you want (just glad to help out)

Answer 36

Someone already gave me a medal - I vote for ranga

Answer 37

wow, i totally missed out on the party..

Answer 38

shamil: you did not get the original formula wrong. If the common ratio is greater than 1 I use a(r^n - 1) / (r-1).
If it is less than 1 I use a(1 - r^n) / (1 - r).
But it does not matter either both the top and bottom will be positive or both will be negative but the result will be the same.

Answer 39

well, for what it's worth, here are my two cents...
a(r^n - 1) / (r-1) = 1001179.38 [ranga's numbers]
then
a_N = 50000(1.04)^15-1 = 86.553.82
thus
1001179.38 + 86.553.82 = total salary earned over 16 years

Answer 40

Sorry DemolisionWolf yes you helped too.
You have the total correct but I had the final salary as 86,583.82

Answer 41

Yes, @shamil98 take note:

n should have been 16 in the formula (as there are 16 terms):
50000(1 + 1.04 + 1.04^2 + 1.04^3 + .... + 1.04^15) =
50000 * (1.04^16 - 1) / (1.04 - 1) = 1,091,226.56

@DemolisionWolf and @wolf1728: The final year's salary should be 90047.17

In Wolf1728's table it should be: 86,583.82 * 1.04 = 90047.17
There are only 15 table when there should be 16.

In DemolisionWolf's calculation, a_N = 50000(1.04)^15 = 90047.17

And if we add 90047.17 to the previous total of 1001179.38 we get 1,091,226.55

Answer 42

oh ok ok, I was reading it weird, so if I were to run the sequence out to 17 years it would look like this then...
(13, 80,051.61)
(14, 83,253.67)
(15, 86,583.82)
(16, 90,047.17) [year16salary = year15salary*1.04]
(17, 90,047.17) [year17salary = year16salary, because there is no more increase]

where I was thinking: (which is incorrect)
(16, 86,583.82) [year16salary = year15salary] <- but this is wrong because the end of year increase affects year 16.
(17, 86,583.82)


*for reference-
"If your starting salary were $50,000 and you received a 4% increase at the end of every year for 15 years, what would be the total amount, in dollars, you would have earned over the first 16 years that you worked?"