Question

Help please medal and fan!

Answer 1

Assume that
\[
1a_1+2a_2+\cdots+20a_{20}=1,
\]
where the \[a_j\] are real numbers and that these values minimize
\[1a_1^2+2a_2^2+\cdots+20a_{20}^2.\]
Find \[a_{12}\].

Answer 2

I already know that the minimum of \[1a_1^2+2a_2^2+\cdots+na_n^2\] is $$\frac{2}{n(n+1)}$$ if \[
1a_1+2a_2+\cdots+na_n=1,
\]

Answer 3

But I'm not sure how to get $$a_{12}$$

Answer 4

Did you try that formula out ?

Answer 5

Yes I got that minimizing \[1a_1^2+2a_2^2+\cdots+20a_{20}^2\] gives $$\frac{1}{210}$$

Answer 6

lets assume that \[a _{n}=a _{n-1}\]

Answer 7

\[a _{n}=n-1\]

Answer 8

can we solve for a12 now ?

Answer 9

Wait how did you get \[a_n=n-1\]?

Answer 10

no, wait i mistook the equation. Sorry.

Answer 11

@rishavraj