Permutations and Combinations

Permutations and Combinations

Question

The sum of two numbers is space 13 over 6. An even number of arithmetic means are being inserted between them and their sum exceeds their number by 1. Find the number of means inserted.

Answer

Let a and b be two numbers such that
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Let space straight A subscript 1 comma space straight A subscript 2 comma space straight A subscript 3........ comma space straight A subscript 2 straight n end subscript be 2n (even) A.M.'s between a and b.

∴   straight A subscript 1 plus straight A subscript 2 plus straight A subscript 3 plus....... plus straight A subscript 2 straight n end subscript space equals space 2 straight n open parentheses fraction numerator straight a plus straight b over denominator 2 end fraction close parentheses space equals space straight n left parenthesis straight a plus straight b right parenthesis               (By using example 4)
                                          = 13 over 6 straight n
Also,            straight A subscript 1 plus straight A subscript 2 plus straight A subscript 3 plus..... plus straight A subscript 2 straight n end subscript space equals space 2 straight n space plus space 1

∴                          2 straight n plus 1 space equals space 13 over 6 straight n    or                12n + 6 = 13n          or        n = 6
Hence,  12 A.M.'s are inserted.

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