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Question
If a, b, c are pth, qth and rth terms of an A.P, find the value of `|("a", "b", "c"),("p", "q", "r"),(1, 1, 1)|`
Solution
Given a, b, c are pth, qth and rth terms of an A.P.
tp = a = A + (p – 1)D,
tq = b = A + (q – 1)D,
tr = 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
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