A community for students.
Here's the question you clicked on:
 0 viewing
zmudz
 one year ago
Assume that
\(
1a_1+2a_2+\cdots+na_n=1,
\)
where the \(a_j\) are real numbers.
As a function of \(n\), what is the minimum value of
\(1a_1^2+2a_2^2+\cdots+na_n^2?\)
I tried 1/5 and then 5, but none of them work. Please help! Thanks!
zmudz
 one year ago
Assume that \( 1a_1+2a_2+\cdots+na_n=1, \) where the \(a_j\) are real numbers. As a function of \(n\), what is the minimum value of \(1a_1^2+2a_2^2+\cdots+na_n^2?\) I tried 1/5 and then 5, but none of them work. Please help! Thanks!

This Question is Closed

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I'm pretty sure you can use Lagrange multipliers to solve this. Let \(S_1=\sum\limits_{j=1}^nja_j\) and \(S_2=\sum\limits_{j=1}^nj{a_j}^2\). You're looking to minimize \(S_2\) with respect to the constraint \(S_1\). The Lagrangian is given by \[\mathcal{L}(a_1,\ldots,a_n,\lambda)=S_2+\lambda S_1\] Take the gradient and set equal to \(0\): \[\left.\begin{align*} \frac{\partial}{\partial a_1}\left[S_2+\lambda S_1\right]&=2a_1+\lambda\\[2ex] \frac{\partial}{\partial a_2}\left[S_2+\lambda S_1\right]&=4a_2+2\lambda\\[2ex] &\vdots\\ \frac{\partial}{\partial a_n}\left[S_2+\lambda S_1\right]&=2na_n+n\lambda\\[2ex] \frac{\partial}{\partial \lambda}\left[S_2+\lambda S_1\right]&=S_1 \end{align*}\right\}=0\quad(\text{each})\] The first \(n\) equations tell you that \(a_j=\dfrac{\lambda}{2}\). Plugging this into the original constraint, you get \[S_1=\sum_{j=1}^n ja_j=\frac{\lambda}{2}\sum_{j=1}^n j=\frac{\lambda n(n+1)}{4}=1\]or \(\lambda=\dfrac{4}{n(n+1)}\). This means \(S_2\) is minimized when \[a_j=\frac{\lambda}{2}=\frac{2}{n(n+1)}\] Compute the sum: \[\begin{align*}\sum_{j=1}^n j{a_j}^2&=\sum_{j=1}^nj\left(\frac{2}{n(n+1)}\right)^2\\[1ex] &=\frac{4}{n^2(n+1)^2}\sum_{j=1}^nj\\[1ex] &=\frac{4}{n^2(n+1)^2}\cdot\frac{n(n+1)}{2}\\[1ex] &=\frac{2}{n(n+1)}\end{align*}\]
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.