Find the value of x 1 2 1 − x 1 + x 2 2...

shmilybaby4i

shmilybaby4i

Answered

2022-06-27

Find the value of x 1 2 1 x 1 + x 2 2 1 x 2 + . . . + x n 2 1 x n ..
Suppose, x 1 + x 2 + . . . + x n = 1   ( x i R , x i 1 ) and x 1 1 x 1 + x 2 1 x 2 + . . . + x n 1 x n = 1. Find the value of x 1 2 1 x 1 + x 2 2 1 x 2 + . . . + x n 2 1 x n .

Answer & Explanation

candelo6a

candelo6a

Expert

2022-06-28Added 24 answers

x 1 2 1 x 1 + x 2 2 1 x 2 + . . . + x n 2 1 x n = x 1 2 1 x 1 + x 2 2 1 x 2 + . . . + x n 2 1 x n 1 + 1 = x 1 2 x 1 1 x 1 + x 2 2 x 2 1 x 2 + . . . + x n 2 x n 1 x n + 1 = ( x 1 + x 2 + . . + x n ) + 1 = 0
telegrafyx

telegrafyx

Expert

2022-06-29Added 8 answers

Let S be the sum of the series x 1 2 1 x 1 + x 2 2 1 x 2 + + x n 2 1 x n , represented by S = i = 1 n x i 2 1 x i
First, let's do some algebra.
x i 2 1 x i = x i 2 2 x i + 2 x i 1 + 1 1 x i = ( x i 2 2 x i + 1 ) + ( x i 1 ) + x i 1 x i = ( x i 1 ) 2 1 x i + x i 1 1 x i + x i 1 x i = ( 1 x i ) 1 + x i 1 x i = x i 1 x i x i
Now we can simplify our sum.
S = i = 1 n x i 2 1 x i = i = 1 n ( x i 1 x i x i ) = i = 1 n x i 1 x i i = 1 n x i = 1 1 = 0

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