Which job would you choose? (math help)?

Question

-job A: salary of 1$ for the first year, 2$ for the second year,4$ for the third year and so on for 25 years

- job B: salary of 1 million dollars for 25 years

1.) At what year will the salaries be closest?

2.) Which job earns the most money over a period of 25 years?

I think I came up with the answer but not sure need a second opinion

could ya'll tell me how you got the answer

Answer 1

Clearly Job B is better.

Surprised ? ... Read on!

Sometimes there is more to these problems than meets the eye!

Or the math!

Let's compare the total amount paid:

Job B pays you $25 million, that's pretty easy.

What about A ?

Sum of powers of 2 = next power of 2 - 1

2^0 + ... + 2^24 = 2^25 - 1

2^20 = 1048576  <-- closest in year 21

2^5 = 32

So pay from job A is 32 * 1048576 which is 33554432.

Oh - forgot to subtract 1. 33554431.

That is more than the 25 million from Job B.

So now you're feeling all smart because you chose Job A.

But not ...  so ... fast ... don't forget about the time value of money!

Let's say your living expenses were $100,000 per year

(think you can get by on that ?)

Job A doesn't deliver big bucks for quite some time.

It takes you 12 years just to get up to $4096/year.

Half the money comes in just the last year.

What were you planning on living on those first years anyway ?

Meanwhile, with Job B, you can save and invest $900,000 per year

waiting for the salary on A to ratchet up.

Let's say you can earn 5 percent on your investment.

By compounding the interest on your $900,000/year savings,

you rack up an additional $40 million in gains there.

Now which job looks better ?

Here is what you make on each year's savings, as

that lowly 5 percent compounds away.

Years Invested | Multiplier | Final amount

1   1   900000  <- no gain on last year's savings

2   1.05        945000 <- gain on next to last year's savings

3   1.1025      992250 and so on

4   1.157625    1041862.5

5   1.21550625  1093955.625

6   1.276281563 1148653.406

7   1.340095641 1206086.077

8   1.407100423 1266390.38

9   1.477455444 1329709.899

10   1.551328216        1396195.394

11   1.628894627        1466005.164

12   1.710339358        1539305.422

13   1.795856326        1616270.693

14   1.885649142        1697084.228

15   1.979931599        1781938.439

16   2.078928179        1871035.361

17   2.182874588        1964587.13

18   2.292018318        2062816.486

19   2.406619234        2165957.31

20   2.526950195        2274255.176

21   2.653297705        2387967.935

22   2.78596259 2507366.331

23   2.92526072 2632734.648 <- 23 years worth on 2nd year's savings

24   3.071523756        2764371.38  <- 24 years worth on first year's savings

Total: 40,051,798.99