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Q.

For every integer n, let $a_{n}$ and $b_{n}$ be real numbers.
Let $f : R \to R$ be given by
$f (x) = {\begin{matrix} a_{n} + \sin π x & , f o r & x \in [2 n, 2 n + 1] \\ b_{n} + \cos π x & , f o r & x \in (2 n - 1, 2 n) \end{matrix},$ for all integers n.
If f is continuous, then which of the following does not hold for all n.

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a

$a_{n} - b_{n} = 1$

b

$a_{n - 1} - b_{n - 1} = 0$

c

$a_{n} - a_{n + 1} = - 2$

d

$a_{n} - b_{n + 1} = - 1$

answer is A.

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Detailed Solution

$a_{n} - b_{n} = 1$
$a_{n} - b_{n + 1} = - 1$
$a_{n - 1} - b_{n} = - 1$
$a_{n - 1} = a_{n + 1} + 1 \Rightarrow a_{n} - a_{n - 1} = 2$

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For every integer n, let an and bn be real numbers. Let f:R→R be given byf(x)={an+sinπx, forx∈[2n,2n+1]bn+cosπx, forx∈(2n−1,2n), for all integers n.If f is continuous, then which of the following does not hold for all n.