Akash Singha Roy

Problem :

Let \(x_1 , x_2, ….. , x_{100} \) be positive integers such that \(x_i + x_{i+1} = k \) for all \(i \) where \(k \) is constant. If \(x_{10} = 1, \) then the value of \(x_1 \) is

(A) \(k \)

(B) \(k – 1 \)

(C) \(k + 1 \)

(D) \(1 \)

Solution:

We have

$LATEX x_i + x_{i+1} = k $ for all \(i \)

Putting \(i = 1, 2, … , 99 \) in the above relation we obtain,

\(x_1 + x_2 = x_2 + x_3 = x_3 + x _ 4 = ……. = x_{99} + x_{100} = k \)

This gives,

\(x_1 = x_3 = x_5 = ……. = x_{99} \)

and

\(x_2 = x_4 = x_6 = ……. = x_{100} \)

Thus, \(x_2 = x_{10} = 1 \)

Now, since \(x_1 + x_2 = k \)

therefore we have,

\(x_1 + 1 = k \)

which, in turn, gives,

\(x_1 = k – 1 \) .

Therefore, option (B) is the correct option.

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