Increasing annuity problem FM Consider a situation where you took a loan for $\$15,000$ and you are paying back the amount at annual effective interest rate of $3\%$, $\textrm{# of periods = 25}$.
Now, you make the first payment of $100$ and then the next payment is $200$. Notice that the payments are increasing by $100$, making it very obvious that we need to use the increasing annuity formula.
Adding to that, once 10 payments are made amounting to a $1000$ dollars, there is an adjustment and the remaining $15$ payments are $\$X$ per year at the same interest rate.
Find $\$X$.
The formula I used in the case is:
$15,000 = 100$ (Ia)for 10 periods + X($v^{10}$) annuity for 15 periods.
$15,000=4,483.8992+X\cdot (1.03)^{-10}\cdot \frac{1-(1.03)^{-15}}{0.03}$
What I am confused about is that why do we need use $v^{10}$ for the remaining $15$ periods ?
The formula is correct and gets to the correct answer of $1183.85$
Note: The payments are being made at the end of each year.

The $15000$ dollars are in present value. When you sum the payments, you want all of them to be in present value, so you want to take the present value of $X$.
$$PV=FVv^{t}$$
And you know $t=10$.
