Qu:
An amount of $3000 is invested in a superannuation fund at the start of each year for 18 years. The interest at a rate of 9% p.a. is paid half-year.
How much is the investment worth after 18 years?
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I don't understand how you need to adjust the equation for the 9% interest being paid every half a year...
You are trying to find future value of an annuity due. The plain formula for this is
FVAD = R x FVIFAD(i%,n)
or
FVAD = R x (1+i) [(1+i)^n - 1]/i
But since we have semiannual compounding we divide i by 2 and multiply n by 2
FVAD = R x (1+i/2) [(1+i/2)^2n - 1]/(i/2)
FVAD = 3000 x (1+9%/2) [(1+9%/2)^36 - 1]/(9%/2)
FVAD = 3000 x (1+4.5%) [(1+4.5%)^36 - 1]/(4.5%)
FVAD = 3000 x (1.045) [(1.045)^36 - 1]/(0.045)
FVAD = 3000 x (1.045) [(1.045)^36 - 1]/(0.045)
FVAD = 3000 x (1.045) [4.8773784614756470522995531708692-1]/0.045
FVAD = 3000 x (1.045) [3.8773784614756470522995531708692]/0.045
FVAD = 3000 x (1.045) [3.8773784614756470522995531708692]/0.045
FVAD = 3000 x (1.045) 86.163965810569934495545626019316
FVAD = 3000 x 90.041344272045581547845179190185
FVAD = $270,124
To see how I got the 90.04 for FVIFAD try this online calculator FVIFAD > Download FVIFAD Table > Online FVIFAD Calculator where you enter 4.5% for interest rate and 36 for periods
To see how to get the FVAD value of $270,124, try this online calculator Online Calculator for Future Value of an annuity due where you enter 3000 for amount and the interest rate as 4.5% and 36 for period