Permutations and Combinations

Permutations and Combinations

Question

In an A.P., the mth term is 1 over straight n and the nth term is 1 over straight m comma find the (mn)th term.

Answer

Let a be the first term and d be the c.d.
             space space space space space space straight t subscript straight m space equals space 1 over straight n space space space space space space space space space rightwards double arrow space space space space space space space space space space space straight a space plus space left parenthesis straight m space minus space 1 right parenthesis straight d space equals space 1 over straight n                                          ...(i)
                 space space straight t subscript straight n space equals space 1 over straight m            rightwards double arrow space space straight a space plus space left parenthesis straight n space minus space 1 right parenthesis straight d space equals space 1 over straight m                                          ... (ii)
Subtracting (ii) from (i), we get
         (m - 1)d - (n - 1)d = 1 over straight n minus space 1 over straight m  or    (m - 1 - n + 1)d = fraction numerator straight m minus straight n over denominator mn end fraction
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Using this value of d in (i), we get
             straight a plus left parenthesis straight m minus 1 right parenthesis 1 over mn space equals space 1 over straight n      or        space space straight a plus straight m over mn minus 1 over mn space equals space 1 over straight n
or        straight a plus 1 over straight n minus 1 over mn equals 1 over straight n      or                             straight a space equals space 1 over mn
                     space space straight t subscript mn space equals space straight a plus space left parenthesis mn minus 1 right parenthesis straight d space equals space 1 over mn plus left parenthesis mn minus 1 right parenthesis 1 over mn space equals space fraction numerator 1 plus mn minus 1 over denominator mn end fraction space equals space mn over mn space equals space 1

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