Maths gurus: QQ

Axe_meister

I want to calculate compound interest, whilst every year putting the same amount into an account.
Let's say I put 1000 into an account every year for 5 years at 4% n=number of years to be held in the savings account.
The formulae would be X*1.04^n+ X*1.04^n-1... until the power =1
Is there anyway to simplify the equation?

Sporky

If you just want the answer there are a load of online calculators - such as (no affiliation):

https://www.thecalculatorsite.com/finance/calculators/compoundinterestcalculator.php

I'd do it with Excel because that's how I do most things.

relic245

Do you just want the answer or are you actually interested in the maths?

If it's the former then this would help:

Compound Interest Calculator - Daily, Monthly, Yearly Compounding (thecalculatorsite.com)

mart

X*1.04^n+ X*1.04^n-1 + ... + X*1.04 is equal to X*(1.04^n+ 1.04^n-1 + ... + 1.04), and the stuff inside the brackets is a geometric series, for which there is a formula: 

If you multiply the bracket stuff by 1.04 then every exponent increases by 1, so if you multiply the bracket stuff by 1.04 - 1, then you get 1.04^(n+1) - 1.04 and all the intermediate powers cancel out.

In other words, 1.04^n+ 1.04^n-1 + ... + 1.04 is equal to (1.04^(n+1) - 1.04) / (1.04-1). And you can even calculate 1.04 - 1 = 0.04, to simplify it further. So your original total becomes 

X*(1.04^(n+1) - 1.04) / 0.04.

Axe_meister

Was actually interested in the maths thanks.

NCo

ChatGPT has this to say:

Yes, the scenario you're describing is a common one in finance, often related to calculating the future value of a series of equal, annual deposits or investments, known as an annuity. The formula you're looking for is for the future value of an annuity compounded annually. The general formula to calculate the future value of such an annuity is:

FV=P×((1+r)n−1r)FV=P×(r(1+r)n−1​)

Where:

    FVFV is the future value of the annuity.
    PP is the annual payment (the amount you deposit every year).
    rr is the annual interest rate (expressed as a decimal).
    nn is the number of years the money is deposited or invested for.

In your case:

    P=1000P=1000
    r=0.04r=0.04 (4% annual interest rate)
    n=5n=5 years

Plugging these values into the formula gives:

FV=1000×((1+0.04)5−10.04)FV=1000×(0.04(1+0.04)5−1​)

FV=1000×(1.2166529−10.04)FV=1000×(0.041.2166529−1​)

FV=1000×(0.21665290.04)FV=1000×(0.040.2166529​)

FV=1000×54.163225FV=1000×54.163225

FV=5416.3225FV=5416.3225

Therefore, after 5 years, the future value of your annuity, with a $1000 annual deposit at an annual interest rate of 4%, would be approximately $5416.32. This formula simplifies the calculation by summing the compounded value of each annual deposit into a single equation, rather than calculating each year's compound interest individually and summing them.

Revolutions

The answer is 42.

stickyfiddle

This is how a mathematician would write it down. 

Where x equals your annual deposit, y equals the interest rate, and n is the number of years