On November 1, 2014, a research fund of $300 000 was established to provide for equal annual grants for 10 years. What will be the size of each grant if
a) the fund earns interest at 8% compounded daily and the first grant is awarded on November 1, 2014?
$$\begin{align} A &= 300000 \\ n &= 10 \\ i &= 0.08/365 = 0.000219178 \\ R &= \frac{300000}{\frac{1-(1.000219178)^{-10}}{0.000219178}} \end{align}$$
$R = 30036.18$ (not the right answer).
b) the fund earns interest at 10% compounded monthly and the first grant is awarded on November 1, 2017?
$$\begin{align} A &= 300000 \\ n &= 7 \\ i &= 0.10/12 = 0.008333333 \\ R &= \frac{300000}{\frac{1-(1.008333333)^{-7}}{0.008333333}} \end{align}$$
$R = 44297.57$ (not the right answer).
What am I doing wrong? Please help
Finance is not my field by a long stretch, but here's how I'd solve Part (a) of the problem. Let $A$ be the amount in the research fund at inception and $R$ the amount of each grant. Since the first grant is immediately paid out, the whole thing is over in $9$ years from the start date.
The effective annual interest rate, given the daily compounding, is
$$ a = \left(1 + \frac{0.08}{365}\right)^{\!365} - 1 = 0.083278 \enspace. $$
The capital plus the interest earned by $A$ over $9$ years is $A(1+a)^9$. However, we immediately pay out one grant. Hence we need to subtract $R(1+a)^9$ to account for the lost interest. After a year, we withdraw another $R$, which would have earned interest for $8$ years; so, we subtract $R(1+a)^8$. And so on. At the end of the ninth year, enough money should be left to pay the tenth grant. Therefore,
$$ A(1+a)^9 - R\,\sum_{1 \leq i \leq 9} (1+a)^i = R \enspace, $$
or, equivalently,
$$ A(1+a)^9 = R\,\sum_{0 \leq i \leq 9} (1+a)^i \enspace, $$
which after a bit of algebra yields
$$ R = \frac{Aa(1+a)^9}{(1+a)^{10}-1} = \frac{a}{1+a} \cdot \frac{A}{1-(1+a)^{-10}} = 41884.02 \enspace. $$
For Part (b), we need a different effective annual rate, $a=0.10471$, and we need to multiply $A$ by $(1+a)^{12}$ instead of $(1+a)^9$ to account for the first three years without payments. If I'm not mistaken, each grant is $60795.57$.
Looking at the formula you used for Part (a), two differences stand out. First, you used the daily interest rate instead of the effective annual rate, which is much higher. Second, it looks like you did your computation as if the first grant were given after one year and the last grant after 10 years. That would indeed multiply $R$ by $(1+a)$.