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For each of the following sequences, • Give a formula for the nth term in the sequence, • Give a recursive definition for the sequence (ie. initial values and a recursive equation). (a) 1, 2, 3, 4, 5, . . . (b) 1, 2, 4, 8, 16, 32, . . . (c) 1, 2, 6, 24, 120, . . .

Answer :

MechEngineer

Answer:

(a)[tex]n^{th} = n[/tex]

[tex]f(1) = 1[/tex]

[tex]f(n) = f(n-1) + 1[/tex]

(b)[tex]n^{th} = 2^{n-1}[/tex]

[tex]f(1) = 1[/tex]

[tex]f(n) = f(n-1) * 2[/tex]

(c)[tex]n^{th} = n![/tex]

[tex]f(1) = 1[/tex]

[tex]f(n) = f(n-1) * n[/tex]

Step-by-step explanation:

(a) This is a sequence of consecutive number

[tex]n^{th} = n[/tex]

[tex]f(1) = 1[/tex]

[tex]f(n) = f(n-1) + 1[/tex]

(b) This is a sequence of 2 to the power of n - 1. The next number is twice time of this number

[tex]n^{th} = 2^{n-1}[/tex]

[tex]f(1) = 1[/tex]

[tex]f(n) = f(n-1) * 2[/tex]

(c) This is factorial sequence. Where the next number is this number multiplied by [tex]n^{th}[/tex]

[tex]n^{th} = n![/tex]

[tex]f(1) = 1[/tex]

[tex]f(n) = f(n-1) * n[/tex]

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