Permutations and Combinations

Permutations and Combinations

Question

Find all the sequences which are simultaneously in A.P. and GP.

Answer

Let the sequence space space open curly brackets straight t subscript straight n close curly brackets  be in A.P. as well as in G.P.
Let straight t subscript straight n comma space space straight t subscript straight n plus 1 end subscript comma space straight t subscript straight n plus 2 end subscript be three consecutive terms of an A.P.

∴            straight t subscript straight n plus 1 end subscript space minus straight t subscript straight n space equals space straight t subscript straight n plus 2 end subscript minus straight t subscript straight n plus 1 end subscript space horizontal strike logical or space straight n greater or equal than 1
or            space space 2 space straight t subscript straight n plus 1 end subscript space equals space straight t subscript straight n space end subscript plus space straight t subscript straight n plus 2 end subscript space horizontal strike logical or space straight n greater or equal than 1                        ...(i)
Let r be the common ratio of G.P.

∴      straight t subscript straight n plus 1 end subscript over straight t subscript straight n space equals space fraction numerator straight t subscript straight n plus 2 end subscript over denominator straight t subscript straight n plus 1 end fraction space equals space straight r
rightwards double arrow                  straight t subscript straight n plus 1 end subscript space equals space straight t subscript straight n straight r  and space space straight t subscript straight n plus 2 end subscript space equals space straight t subscript straight n plus 1 straight r space equals space straight t subscript straight n straight r squared

Using the values of straight t subscript straight n plus 1 end subscript and space space straight t subscript straight n plus 2 end subscript in (i), we get
                   2 straight t subscript straight n straight r space equals space straight t subscript straight n space plus space straight t subscript straight n straight r squared              rightwards double arrow space space space space space space space space space space space space space space space space space space space space space 2 straight r space equals space 1 plus straight r squared
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#6 {main}</pre>                         rightwards double arrow            left parenthesis straight r minus 1 right parenthesis squared space equals space 0
rightwards double arrow                  r = 1

∴                     space space straight t subscript 1 space equals space straight t subscript 2 space equals space straight t subscript 3 equals space.............          
Hence, all the constant sequences are the only sequences which are in A.P. as well as in G.P.

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