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 $latex x_1 , x_2, ..... , x_{100} $ be positive integers such that $latex x_i + x_{i+1} = k $ for all $latex i $ where $latex k $ is constant. If $latex x_{10} = 1, $ then the value of $latex x_1 $ is
(A) $latex k $
(B) $latex k - 1 $
(C) $latex k + 1 $
(D) $latex 1 $
Solution:
We have
$LATEX x_i + x_{i+1} = k $ for all $latex i $
Putting $latex i = 1, 2, ... , 99 $ in the above relation we obtain,
$latex x_1 + x_2 = x_2 + x_3 = x_3 + x _ 4 = ....... = x_{99} + x_{100} = k $
This gives,
$latex x_1 = x_3 = x_5 = ....... = x_{99} $
and
$latex x_2 = x_4 = x_6 = ....... = x_{100} $
Thus, $latex x_2 = x_{10} = 1 $
Now, since $latex x_1 + x_2 = k $
therefore we have,
$latex x_1 + 1 = k $
which, in turn, gives,
$latex x_1 = k - 1 $ .
Therefore, option (B) is the correct option.
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