For all arithmetic sequences $a_1, a_2, a_3,...$ that has $(a_1)^2+(a_{11})^2\leq100$, what's the maximum value of $S=a_{11}+a_{12}+...+a_{21}$? For all arithmetic sequences $a_1, a_2, a_3,...$ that has $(a_1)^2+(a_{11})^2\leq100$, what's the maximum value of $S=a_{11}+a_{12}+...+a_{21}$?
I was given this question for homework for a class I'm taking, and I don't know how  to start on it. Can someone help me out?
 A: If the difference of the sequence is $d,$ then you know that $a_1^2 + (a_1+ 10d )^2 \leq 100,$ and you want to find the maximum of $11a_1 + 165 d$ (check the arithmetic). Lagrange multipliers are your friend here.
EDIT To avoid Lagrange multipliers:


*

*write $a_1^2 + (a_1+10d)^2$ as an ellipse in standard form (completing various squares). 

*Change variables to make the ellipse a disk

*Find the intersection of the line (in new coordinates) with the disk. There will be two points, one of which will have the bigger value.
A: One way to start:
$$
 S = a_{11}+a_{12}+\cdots+a_{21} = \frac{11}2(a_{11} + a_{21}),
$$
$$
a_{21} = a_1 + 20\left(\frac{a_{11}-a_1}{10}\right)
= 2 a_{11} -  a_1.
$$
$$
S = \frac{11}2(3a_{11} - a_1).
$$
So you just need to maximize $\frac{11}2(3a_{11} - a_1)$
with the constraint $a_1^2 + a_{11}^2 \leq 100.$
That's the same as maximizing $\frac2{11}S = 3a_{11} - a_1$
under the same conditions.
It may also help if  you determine that when $S$ is maximized,
$a_{11} > 0$ and $a_1^2 + a_{11}^2 = 100.$
