Permutations and Combinations

Permutations and Combinations

Question

If m times the mth term is equal to n times the nth term of an A.P. Prove that (m + w)th term of an A.P. is zero.

Answer

Let a be the first term and d be the common difference
Since m times the mth term is equal to n times the nth term.

∴                                mt subscript straight m space equals space nt subscript straight n
or                      m[a + (m - 1)d] = n[a + (n - 1)d]
or                      ma + m(m - 1)d = na + n (n-1)d
or       ma - na + left parenthesis straight m squared minus straight m right parenthesis straight d space minus space left parenthesis straight n squared minus straight n right parenthesis straight d space equals space 0
or           (m - n) a + (straight m squared minus straight m minus straight n squared plus straight n right parenthesis straight d space equals space 0
or      (m - n) a + [(m2 - n2) - (m - n)] d = 0
or       (m - n) a + [(m - n) (m + n) - (m - n)]d = 0
or             (m - n) a + (m - n) (m + n - 1)d = 0
or               (m - n) [a + (m + n - 1)d] = 0
or                        a + (m + n - 1) d =0                                   (∵   straight m not equal to straight n)
Hence,                            straight t subscript straight m plus straight n end subscript space equals space 0 

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