Let a- b- c be such that b-c - 0- If- a a-1 a-1- -b b-1 b-1- c c-1 c-1 - - a-1 b-1 c-1- a-1 b-1 c-1- -1n-2 a -1n-1 b -1n c - -0Then- the value of -n- is

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Properties of Determinants

Let a, b, c b...

Question

Let a, b, c be such that (b+c) ≠0. If

Then, the value of 'n' is

Zero

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An even integer

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An odd integer

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Any integer

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Solution

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(a + c) \neq 0

∣ ∣ ∣ ∣ \begin{matrix} a & a + 1 & a - 1 - b & b + 1 & b - 1 c & c - 1 & c + 1 \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} a + 1 & b + 1 & c - 1 a - 1 & b - 1 & c + 1 (- 1)^{n + 2} a & (- 1)^{n + 1} b & (- 1)^{n} c \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

a, b, c

b (a + c) \neq 0

∣ ∣ ∣ ∣ \begin{matrix} a & a + 1 & a - 1 - b & b + 1 & b - 1 c & c - 1 & c + 1 \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} a + 1 & b + 1 & c - 1 a - 1 & b - 1 & c + 1 (- 1)^{n + 2} a & (- 1)^{n + 1} b & (- 1)^{n} c \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0,

n

∣ ∣ ∣ ∣ \begin{matrix} a & a + 1 & a - 1 - b & b + 1 & b - 1 c & c - 1 & c + 1 \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} a + 1 & b + 1 & c - 1 a - 1 & b - 1 & c + 1 {(- 1)}^{n + 2} a & {(- 1)}^{n + 2} b & {(- 1)}^{n} c \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

n

∣ ∣ ∣ ∣ ∣ \begin{matrix} (- 1)^{n} a & (- 1)^{n + 1} b & (- 1)^{n + 1} c a + 1 & 1 - b & 1 + c a - 1 & b + 1 & 1 - c \end{matrix} ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} (- 1)^{n + 1} a & a + 1 & a - 1 (- 1)^{n} b & 1 - b & b + 1 (- 1)^{n + 2} c & 1 + c & 1 - c \end{matrix} ∣ ∣ ∣ ∣ ∣

\frac{a + b + c}{a^{- 1} b^{- 1} + b^{- 1} c^{- 1} + a^{- 1} c^{- 1}} = a b c

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