Permutations and Combinations

Permutations and Combinations

Question

Find the first six terms of the sequence whose first term is 1 and whose (n+l)th term is obtained by adding n to the nth term.

Answer

Let  straight t subscript straight n denote the nth term
               straight t subscript 1 space end subscript space equals space 1
According to given equation  straight t subscript straight n plus 1 end subscript space equals space straight t subscript straight n plus straight n                        ...(i)
Putting n = 1 in (i), we get          straight t subscript 1 plus 1 end subscript space equals space straight t subscript 1 plus 1 space equals space 1 plus 1 space equals space 2      rightwards double arrow space straight t subscript 2 space equals space 2
Putting n = 2 in (i), we get          straight t subscript 2 plus 1 end subscript space equals space straight t subscript 2 plus 2 space equals space 2 plus 2 space equals space 4    space space rightwards double arrow space space straight t subscript 3 space equals space 4
Putting n = 3 in (i), we get          straight t subscript 3 plus 1 end subscript space equals space straight t subscript 3 plus 3 space equals space 4 plus 3 equals 7      space space rightwards double arrow space space space space straight t subscript 4 space equals space 7 
Putting n = 4 in (i), we get         straight t subscript 4 plus 1 end subscript space equals space straight t subscript 4 plus 4 space equals space 7 space plus space 4 space equals space 11    rightwards double arrow space straight t subscript 5 equals 11
Putting n = 5 in (i), we get        straight t subscript 5 plus 1 end subscript space equals space straight t subscript 5 space plus space 5 space equals space 11 space plus space 5 space equals space 16   rightwards double arrow space straight t subscript 6 space equals space 16

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