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?#

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Answer 1 · 2017-05-21T00:31:23

#2/(n(n+1)#

#sum_(k=1)^n k a_k = 1# is the restriction and #sum_(k=1)^n k a_k^2# is the objective function

The Lagrangian reads

#L(a,lambda)=sum_(k=1)^n k a_k^2+lambda(sum_(k=1)^n k a_k - 1) #

The stationary conditions

#(partialL)/(partial a_k) = 2ka_k+lambda k = 0# for #k=1,2,cdots,n#

and

#(partialL)/(partial lambda) = sum_(k=1)^n k a_k- 1 = 0#

so

#a_k = -lambda/2# and

#sum_(k=1)^n k ( -lambda/2)= 1 # or

#lambda = -4/(n(n+1))# so

#a_k = 2/(n(n+1))#

so the minimum value is given by

#sum_(k=1)^n k (4/(n(n+1))^2) = 2/(n(n+1))#