I am working on the following problem and I keep getting a different answer.
The principal $P=10,000$. The annual interest rate is $i=4\%$. The money is deposited at time 0 and the interest is compounded yearly for 10 yrs at the end of each year. Four withdrawals with amount $K$ are made at time at the end of the 4th, 5th, 6th and 7th year. When a withdrawal is made there is a 5% penalty. The amount at time 10 is $10,000$. Find $K$.
The following is my solution.
I am thinking that the deposit is made at $t=0$ with the amount $P$ so after 10 yrs it would accumulate to $P(1.04)^{10}$.
The amount taken from the account at year 4, i.e. when $t=4$, including the penalty is $K(1.05)$, which loss accumulates to $K(1.05)(1.04)^6$ by time $t=10$.
A similar argument can be made with the withdrawals made at time $t=5,6,7$.
Finally, since the total accumulated amount is the same as the original value, we have the equation
$$P(1.04)^{10}-K(1.05)((1.04)^6+(1.04)^5+(1.04)^4+(1.04)^3)=P$$
When I solve for $K$ I get the answer 957.51 but the answer is supposedly 979.93 .
Can someone see where my argument is wrong?