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 2 straight n squared plus straight n plus 1 is not an A.P.

Answer

Here,  straight t subscript straight n space equals space 2 straight n squared plus straight n plus 1
Replacing by n - 1, we get
                     straight t subscript straight n minus 1 end subscript space equals space 2 left parenthesis straight n minus 1 right parenthesis squared plus left parenthesis straight n minus 1 right parenthesis space plus 1 space equals space 2 left parenthesis straight n squared space minus space 2 straight n space plus space 1 right parenthesis space plus space straight n minus 1 space plus 1
                             equals 2 straight n squared minus 4 straight n plus 2 plus straight n space equals space 2 straight n squared minus 3 straight n plus 2equals 2 straight n squared minus 4 straight n plus 2 plus straight n space equals space 2 straight n squared minus 3 straight n plus 2
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#6 {main}</pre>
                           equals 2 straight n squared plus straight n plus 1 minus 2 straight n squared plus 3 straight n minus 2 space equals space 4 straight n minus 1
(which depends upon n and is therefore not a constant)
Hence, the given sequence is not an A.P.

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