Each year on her birthday, Jane’s parents put \$20 into an investment account earning $9\frac{1}{2}$% per annum compound interest. The first deposit took place on the day of her birth. On her 18th birthday, Jane’s parents gave her the account and \$20 cash in hand. How much money did she receive from her parents on her 18th birthday?
So $$A_n = 20\frac{1.095^n-1}{0.095}$$
However, I am unsure what to make of the twenty cash on hand. I think it is 20 dollars on top of the account so $A_{18} + 20$. However the answers suggest simply $A_{19}=970$ but then that is $20$ including interest on the final 20 right? $A_{19}$ factors in the "20 dollars cash in hand" but since $A_0 = 20(1.095)$, that final "20 dollars cash in hand" would have interest applied to it which doesn't seem implied in the question.
I am unsure at this stage
In this annuity calculation, the accumulated value of the annuity is to be calculated on the date of the last payment. That makes it an ordinary annuity, as opposed to an annuity due, where the value is found one period after the last payment.
So just use the formula for an the accumulated value of an ordinary annuity lasting 19 years with 19 annual payments of $20, the first made one year after the annuity "starts", and the last made 19 years after the annuity "starts".
The framers of the question confused things by having payments made on the exact dates of the first and last payments. There is no standard formula for such a scheme although one could be created quite easily.
It may seem contradictory to consider the annuity "starts" a year before Jane was born (or even conceived!). But in. for example, a home mortgage, you take out a loan, receive the amount of the loan (!) and the mortgage starts, but you don't make the first payment until one period has passed.