Chuck needs to purchase an item in $10$ years. The item costs $ \$200$ today, but its price inflates at $4 \%$ per year. To finance the purchase, Chuck deposits $ \$20$ into an account at the beginning of each year for $6$ years. He deposits an additional $X$ at the beginning of years $4, 5,$ and $6$ to meet his goal. The annual effective interest rate is $10 \%$. Calculate $X$.
This is how i interpret the problem: You have $5$ cash flows starting from $0$ to $5$ of $ \$ 20$. You also have $3$ cash flows at $t=4,5,6$.
I used annuity due formula to shift former cash flow to year $6$, and then accumulate it to year $10$ by the $4$ remaining years.
I used the same approach for the latter:
$(20\ddot{s}_{\overline{5|}i=10\%})(1.1)^4 + (X \ddot{s}_{\overline{3|}i=10\%})(1.1)^3 = 200(1.04)^{10}\tag{1}$
But this does not give me the right answer. Can someone please tell me what I'm doing wrong? Thanks in advance.
Each time Chuck puts some money on the bank account, all you need to do is capitalize this amount until maturity. First deposit of 20 will give him $20 * (1.1)^{10}$ at maturity. Second one $20 * (1.1)^{9}$. So the 6 known deposits are treated easily. Then you do the same for the unknown deposits 'X'. A X deposit at beginning of year 4 is capitalized into $x * (1.1)^{7}$ at maturity. It is the same idea for deposits at year 5 and 6.
Then the total of the capitalized deposits must be equal to $200 * (1.04)^{10}$. Which should give you X easily.