MathDB

Let

a_1,a_2, \dots a_n

be positive real numbers. Define

b_1,b_2, \dots b_n

as follows. \begin{align*} b_1&=a_1 \\ b_2&=max(a_1,a_2)\\ b_i&=max(b_{i-1},b_{i-2}+a_i) \text{ for } i=3,4 \dots n \end{align*} Also define

c_1,c_2 \dots c_n

as follows. \begin{align*} c_n&=a_n \\ c_{n-1}&=max(a_n,a_{n-1})\\ c_i&=max(c_{i+1},c_{i+2}+a_i) \text{ for } i=n-2,n-3 \dots 1 \end{align*} Prove that

b_n=c_1

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Problem Statement