Question

If a, b, c, are all positive, and are p^th, q^th and r^th terms of a G.P., show that `|(log"a", "p", 1),(log"b", "q", 1),(log"c", "r", 1)|` = 0

Answer 1

Given a, b, c are the p^th, q^th and r^th terms of a G.P.

∴ a = AR^p-1

b = AR^q-1

c = AR^r-1

Where A is the first term, R – common ratio.

Let Δ = `|(log"a", "p", 1),(log"b", "q", 1),(log"c", "r", 1)|`

= `|(log"AR"^("p" - 1)  "p", 1),(log"AR"^("q" - 1)  "q", 1),(log"AR"^("r" - 1)  "r", 1)|`

= `|(log"A" + log"R"^("p" - 1)  "p", 1),(log"A" + log"R"^("q" - 1)  "q", 1),(log"A" + log"R"^("r" - 1)  "r", 1)|`

= `|(log"A", "p", 1),(log"A", "q", 1),(log"A","r", 1)| + |(log"R"^("p" - 1)  "p", 1),(log"R"^("q" - 1)  "q", 1),(log"R"^("r" - 1)  "r", 1)|`

Property 7:

`|("a"_1 + "m"_1, "b"_1, "c"_1),("a"_2 + "m"_2, "b"_2, "c"_2),("a"_3 + "m"_3, "b"_3, "c"_3)| = |("a"_1, "b"_1, "c"_1),("a"_2, "b"_2, "c"_2),("a"_3, "b"_3, "c"_3)| + |("m"_1, "b"_1, "c"_1),("m"_2, "b"_2, "c"_2),("m"_3, "b"_3, "c"_3)|`

= `log"A"|(1, "p", 1),(1, "q", 1),(1, "r", 1)| + |(("p" - 1) log"R", "p", 1),(("q" - 1) log"R", "q", 1),(("r" - 1)log"R", "r", 1)|`

= `log"A" xx 0 + log"R" |("p" - 1, "p", 1),("q" - 1, "q", 1),("r" - 1, "r", 1)|  "C"_1 -> "C"_1 + "C"_3`

Property 4: If two rows (columns) of a determant are identical then its determinant value is zero.

= `0 + log"R"|("p" - 1 + 1, "p", 1),("q" - 1 + 1, "q", 1),("r" - 1 + 1, "r", 1)|`

= `0 + log"R"|("p", "p", 1),("q", "q", 1),("r", "r", 1)|`

Two columns of the determinant are identical.

 = 0 + log R × 0 

= 0