Permutations and Combinations

In an A.P., the mth term is and the nth term is find the (mn)th term. - Wired Faculty

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.

Solution

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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Permutations And Combinations

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.

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Determine K, so that K + 2, 4K – 6 and 3K – 2 are three consecutive terms of an A.P.

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