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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

$x_i + x_{i+1} = k$ 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.