Permutations and Combinations

Permutations and Combinations

Question

Insert 6 arithmetic means between 3 and 24.

Answer

Let straight A subscript 1 comma space space straight A subscript 2 comma space space straight A subscript 3 comma space space straight A subscript 4 comma space straight A subscript 5 comma end subscript space straight A subscript 6 be six A.M.'s between 3 and 24.

∴  3, straight A subscript 1 comma space space straight A subscript 2 comma space space straight A subscript 3 comma space straight A subscript 4 comma space space straight A subscript 5 comma space space straight A subscript 6 comma space space 24 are in A.P.
Here, a =3, Let d be the common difference
   space space straight t subscript 8 space equals space 24 space space space space space space space space rightwards double arrow straight a plus left parenthesis 8 minus 1 right parenthesis straight d space equals space 24 space space rightwards double arrow space space 3 plus 7 straight d space equals space 24 space space space space space space space space space space space space rightwards double arrow space space 7 straight d space equals space 21 space space space space space space space rightwards double arrow space space straight d space equals space 3
                     straight A subscript 1 space equals space straight t subscript 2 space equals space straight a plus straight d equals 3 plus 3 equals 6 comma space space space straight A subscript 2 space equals space straight t subscript 3 space equals space straight a plus 2 straight d space equals space 3 plus 2 left parenthesis 3 right parenthesis space equals space 9
                     straight A subscript 3 space equals space straight t subscript 4 space equals space straight a plus 3 straight d space equals space 3 space plus space 3 left parenthesis 3 right parenthesis space equals space 12 comma space space straight A subscript 4 space equals space straight t subscript 5 space equals space straight a space plus space 4 straight d space equals space 3 space plus space 4 left parenthesis 3 right parenthesis space equals space 15
                   space space space straight A subscript 5 space equals space straight t subscript 6 space equals space straight a plus 5 straight d equals 3 plus 5 left parenthesis 3 right parenthesis space equals space 18 comma space space space straight A subscript 6 space equals straight t subscript 7 equals straight a plus 6 straight d equals 3 plus 6 left parenthesis 3 right parenthesis space equals space 21
∴      Six A.M.'s between 3 and 24 are 6, 9, 12, 15, 18, 21.

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