Question

If a, b, c are p^th, q^th and r^th terms of an A.P, find the value of `|("a", "b", "c"),("p", "q", "r"),(1, 1, 1)|`

Answer 1

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

t_p = a = A + (p – 1)D,

t_q = b = A + (q – 1)D,

t_r = c = A + (r – 1) D

Where A – first term, D – Common difference of the A.P.

`|("a", "b", "c"),("p", "q", "r"),(1, 1, 1)| = |("A" + ("p" - 1)"D", "A" + ("q" - 1)"D", "A" + ("r" - 1)"D"),("p", "q", "r"),(1, 1, 1)|`

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

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

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

= 0 + D × 0   Two rows are same

 `|("a", "b", "c"),("p", "q", "r"),(1, 1, 1)|` = 0