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Question

Let a, b and c be such that b(a+c) 0.
If ∣ ∣aa+1a1bb+1b1cc1c+1∣ ∣+∣ ∣ ∣a+1b+1c1a1b1c+1(1)n+2a(1)n1b(1)nc∣ ∣ ∣=0, then the value of n is

A
zero
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B

any even integer

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C
any odd integer
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D
any integer
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Solution

The correct option is C any odd integer
∣ ∣aa+1a1bb+1b1cc1c+1∣ ∣+(1)n∣ ∣a+1b+1c1a1b1c+1abc∣ ∣
=∣ ∣aa+1a1bb+1b1cc1c+1∣ ∣+(1)n∣ ∣a+1a1ab+1b1bc1c+1c∣ ∣
=∣ ∣aa+1a1bb+1b1cc1c+1∣ ∣+(1)n+2∣ ∣aa+1a1bb+1b1cc1c+1∣ ∣
This is equal to zero only if n + 2 is odd, i.e, n is an odd integer.

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