Permutations and Combinations

Permutations and Combinations

Question

Show that the sequence space space open curly brackets straight t subscript straight n close curly brackets defined by straight t subscript straight n space equals space nA space plus space straight B (where A and B are constant) is an A.P. with common difference A.

Answer

Here,    straight t subscript straight n space equals space nA space plus space straight B
Replacing n by n - 1, we get
         space space straight t subscript straight n minus 1 end subscript space equals space left parenthesis straight n minus 1 right parenthesis straight A space plus space straight B space equals space straight n space straight A space minus space straight A space plus space straight B
            straight d space equals space straight t subscript straight n space minus space straight t subscript straight n minus 1 end subscript space equals space left parenthesis nA space plus space straight B right parenthesis space minus space left parenthesis nA space minus space straight A space plus space straight B right parenthesis space equals space nA space plus space straight B space minus space nA space plus space straight A space minus space straight B space equals space straight A
which is constant and independent of n.
Hence, the sequencespace space open curly brackets straight t subscript straight n close curly brackets  is an A.P.

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