This is a problem from Test of Mathematics, TOMATO problem 6. It is useful for ISI and CMI Entrance Exam.

Problem : TOMATO Problem 6 

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.

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