robin invests $50 on the first of every month into a superannuation account.

His employer, Batman, pays 9% of Robin's monthly income of $2500 at the
end of the month into the same account. Interest is paid at the rate of 6% p.a.
compounded monthly.
(a) Calculate the total amount in the account after 45 years.
(b) How much more should Robin invest each month if he wishes to have a
superannuation fund value of $1 000 000 after 45 years?

Since the common formula for the amount of an annuity :

amount = Payment( (1+i)^n - 1)/i
is base on the fact that the payment is made at the end of the period, the only twist that I see in the question would be the first month and the last month.
I will also assume that the end of one month is equivalent to the beginning of the next months

Beginning of month 1 --- 50
Beginning of month 2 --- 50+225
Beginning of month 3 --- 50+225
.....
Beginning of month 539 --- 50+225
Beginning of month 540 --- 50+225
End of month 540--------- 225

I see it as
50(1.005^540 - 1)/.005 (1.005) + 225(1.005^540 - 1)/.005
= ...

let me know what you get.

To calculate the total amount in Robin's account after 45 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = the total amount in the account after time t
P = the initial investment (monthly contribution)
r = annual interest rate (converted to decimal)
n = number of times interest is compounded per year
t = time in years

Let's calculate part (a) first:

P = $50 (monthly investment)
r = 6% p.a. = 0.06 (annual interest rate)
n = 12 (compounded monthly)
t = 45 (years)

Now we can substitute these values into the formula:

A = $50(1 + 0.06/12)^(12*45)

Calculating this expression will give us the total amount in the account after 45 years.

For part (b), we need to find out how much more Robin should invest each month to reach a superannuation fund value of $1,000,000 after 45 years. To do this, we can set up an equation:

P' = amount Robin needs to invest each month after adjustment
r = 6% p.a. = 0.06 (annual interest rate)
n = 12 (compounded monthly)
t = 45 (years)
A' = $1,000,000 (desired superannuation fund value)

Now we need to solve for P':

1,000,000 = P'(1 + 0.06/12)^(12*45)

We can rearrange this equation to solve for P'.

Now that we have all the necessary equations set up, we can solve for the respective values to get the answers to both parts (a) and (b).