Present value of deferred annuity with varying amounts An annual annuity pays the amount 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 (in dollars), the first payment occurring at the end of the second year. The present value at $=0$ of this annuity is 25 dollars at an annual effective rate $i$.
Another annual annuity pays the amount 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1 (in dollars), the first payment occurring at the end of the first year. The present value at $=0$ of this annuity is $V$ dollars at an annual effective rate $i$.
What is the value of $V$?
I have been trying to do this problem for two days, but I do not know where to start. Do I need to find the annual effective rate first using the present value given? My issue with this is that since the payments are not all equal, do I need to calculate them all individually? Or is there a formula?
Thank you !!

Let $$p(v)=v^2+2v^3+3v^4+4v^5+5v^6+6v^7+5v^8+4v^9+3v^{10}+2v^{11}+v^{12}$$
Since $p(0)=0$, $p$ is strictly increasing on $v\geq0$, and $p(v)\to\infty$ as $v\to\infty$, there is a unique $r>0$ such that $p(r)=25$.
Now, $$\begin{align}
p(v)&=v^2(1+2v+3v^2+4v^3+5v^4+6v^5+5v^6+4v^7+3v^8+2v^9+v^{10})\\
&=v^2(1+v+v^2+v^3+v^4+v^5)^2\\
&=\left(v\frac{v^6-1}{v-1}\right)^2
\end{align}$$
where the second equation is proved in the accepted answer to this question, so that $$\frac{r^7-r}{r-1}=5$$
Let $$q(v)=v+2v^2+3v^3+4v^4+5v^5+6v^6+6v^7+5v^8+4v^9+3v^{10}+2v^{11}+v^{12}$$
Then $$\begin{align}
q(v)-p(v)&=v+2v^2+3v^3+4v^4+5v^5+6v^6-(v^2+2v^3+3v^4+4v^5+5v^6)\\
&=v+v^2+v^3+v^4+v^5+v^6\\&=\frac{v^7-v}{v-1}
\end{align}$$ so that
$$q(r)-p(r)=\frac{r^7-r}{r-1}=5$$ and $$q(r)=25+5=\boxed{30}$$
