Question

Jan's initial investment is $6000. She is going to deposit $500/mo ($6K/yr) for the next 45 years and get 7% compound interest annually. How is the answer $1,778,831? Can you please provide the formula and how this answer comes to this amount?

Answer 1

This sum can be expressed in the formula:
`S = A*[(1+P)^(N+1)-1]/P`
where
`A` - initial and annual investment
`P` - annual interest
`N` - number of years

Explanation:

My first assumption was that the interest is paid annually at `7%` rate. The result for `45` years of accumulation was smaller than your amount.
Much closer result was with semi-annual payment of interest at 3.5% each half a year, which is equivalent to `7.1225%` annually.

Here is the theory.
At year `0` we put amount `A` in the bank for annual interest (paid annually) of `P`.
At year `1` we added the same amount `A` for the same interest `P`.
Do the same up to year `N`.
What is the final sum?

Since the initial amount `A` grows for `N` years, it accumulates into `A*(1+P)^N`.
Additional amount `A` contributed at the end of the first year grows for `N-1` years and accumulates into `A*(1+P)^(N-1)`.
Continue this process for `N` years. The total sum of all investments with interest will be
`A*(1+P)^N + A*(1+P)^(N-1) + A*(1+P)^(N-2) + ... + A*(1+P)^1 + A*(1+P)^0`

This sum can be expressed in the more compact formula using the formula for a sum of geometric progression with factor `(1+P)`:
`S = A*[(1+P)^(N+1)-1]/[(1+P)-P] = A*[(1+P)^(N+1)-1]/P`

For `A=6,000`, `P=0.07` and `N=44` this formula gives
`S=$1,714,496` (not your amount)
For `N=45` the result is
`S=$1,840,511` (also not your amount)

If we assume that the interest is compounded semi-annually (as most bonds do) at the rate `7/2%=3.5%` per half a year, this would result in an annual factor of greater than `1.007`:
`(1+(7/200))^2 = 1.071225`

For `A=6,000`, `P=0.071225` and `N=44` this formula gives
`S=$1,778,491` (almost your amount)
The difference between this and your amount might be attributed to rounding during the calculations.