If a1, a2, ....... an, ..... are in G.P. then is

Engineering

Mathematics

Introduction to Determinants

Geometric Progression and Its Properties

Properties of Logarithm

Question

If a₁, a₂, ....... a_n, ..... are in G.P. then $|\begin{matrix} {loga}_{n} & {loga}_{n + 1} & {loga}_{n + 2} \\ {loga}_{n + 3} & {loga}_{n + 4} & {loga}_{n + 5} \\ {loga}_{n + 6} & {loga}_{n + 7} & {loga}_{n + 8} \end{matrix}|$ is

– 1

None of these

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We know that $\frac{a_{2}}{a_{1}} = \frac{a_{3}}{a_{2}} = .. .. .. . \frac{a_{n}}{a_{n - 1}} = r$

which means that a_n, a_n+1 and a_n+2 are in G.P

So,

a²_n+1 = a_n × a_n+2

Taking log on both the sides, we get

2loga_n+1 = loga_n + loga_n+2

2loga_n+1 – loga_n – loga_n+2 = 0 --------(1)

Similarly,

2loga_n+4 – loga_n+3 – loga_n–5 = 0 -------(2)

2loga_n+7 – loga_n+6 – loga_n+8 = 0 ---------(3)

Apply the operation on column, we get

C₁ → 2C₂ – C₁ – C₃

det(A) = 0