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GO Classes DPP | GATE CS | Discrete Mathematics| Functions

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Given :

  • A recursive function $f$ where $f(0) = 2$

  • The recursive rule is $f(n+1) = 2f(n) + n + 1$ for $n \ge 0$

 

We need to calculate the values sequentially from $f(1)$ up to $f(5)$.

  • For $n = 0$:

    $$f(0+1) = 2f(0) + 0 + 1$$

    $$f(1) = 2(2) + 1 = 5$$

  • For $n = 1$:

    $$f(1+1) = 2f(1) + 1 + 1$$

    $$f(2) = 2(5) + 2 = 12$$

  • For $n = 2$:

    $$f(2+1) = 2f(2) + 2 + 1$$

    $$f(3) = 2(12) + 3 = 27$$

  • For $n = 3$:

    $$f(3+1) = 2f(3) + 3 + 1$$

    $$f(4) = 2(27) + 4 = 58$$

  • For $n = 4$:

    $$f(4+1) = 2f(4) + 4 + 1$$

    $$f(5) = 2(58) + 5 = 121$$

 

Answer: $\boxed{f(5) = 121}$

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