If-a-gt-0-b-gt-0-c-gt-0-are-respectively-the-p-th-q-th-r-th-terms-of-a-GP-then-the-value-of-the-determinant-determinant-log-a-p-1-log-b-q-1-log-c-r-1-is-

Question Number 7911 by ashis786 last updated on 24/Sep/16

If a > 0, b > 0, c > 0 are respectively the

p^{th}, q^{th}, r^{th} terms of a GP, then the value

of the determinant | \begin{matrix} \log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1 \end{matrix} | is

Answered by prakash jain last updated on 24/Sep/16

Let first term of GP be x y with common ratio y

a = {x y}^{p} \Rightarrow \log a = \log x + p \log y

b = {x y}^{q} \Rightarrow \log b = \log x + q \log y

c = {x y}^{r} \Rightarrow \log c = \log x + r \log y

| \begin{matrix} \log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1 \end{matrix} | = | \begin{matrix} \log x + p \log y & p & 1 \\ \log x + q \log y & q & 1 \\ \log x + r \log y & r & 1 \end{matrix} |

= | \begin{matrix} \log x & p & 1 \\ \log x & q & 1 \\ \log x & r & 1 \end{matrix} | + | \begin{matrix} p \log y & p & 1 \\ q \log y & q & 1 \\ r \log y & r & 1 \end{matrix} |

= \log x | \begin{matrix} 1 & p & 1 \\ 1 & q & 1 \\ 1 & r & 1 \end{matrix} | + \log y | \begin{matrix} p & p & 1 \\ q & q & 1 \\ r & r & 1 \end{matrix} |

C 1 = C 3 (in 1^{st} determinant) and C 1 = C 2 in 2^{nd}

= (\log x) \times 0 + \log y \times 0 = 0

Commented by Rasheed Soomro last updated on 24/Sep/16

A l s o l i k e y o u r w a y o f c o r r e c t i o n .

Commented by ashis786 last updated on 24/Sep/16

t h a n k y o u s o m u c h \dots

Commented by sandy_suhendra last updated on 24/Sep/16

i f t h e f i r s t t e r m o f G P = x a n d t h e r a t i o = y

s o a = t h e p^{t h} t e r m o f G P = {x y}^{p - 1}

{i s n}^{'} t i t ?

Commented by prakash jain last updated on 24/Sep/16

Thanks . Updated the answer to start with

first term x y .