text: 1. ## Induction with divisibility
  
  Define f: (N U {0}) -> N by $f(n) = 10^n + 3*4^{n+2}+5$ Prove that for all non-negative numbers, n, f(n) is divisible by 9.
  --------------------
  Ok so for this one P(n) is going to be the statement "f(n) is divisible by 9". P(0) holds and we can now assume that this holds true for all j<k.
  
  This part of induction always gets me, now I have to show P(j) -> P(k), or P(n) -> P(n+1).
  
  If 9 divides f(j), then there is some number, D, such that $9*D = 10^j + 3*4^{j+2}+5$. Now I know there's some way to manipulate this to get what I need but I don't see it.
  
  Hints?
  
  2. Originally Posted by Jameson
  Define f: (N U {0}) -> N by $f(n) = 10^n + 3*4^{n+2}+5$ Prove that for all non-negative numbers, n, f(n) divides 9.
  
  --------------------
  Ok so for this one P(n) is going to be the statement "f(n) divides 9". P(0) holds and we can now assume that this holds true for all j<k.
  
  This part of induction always gets me, now I have to show P(j) -> P(k), or P(n) -> P(n+1).
  
  If f(j) divides 9, then there is some number, D, such that $9*D = 10^j + 3*4^{j+2}+5$. Now I know there's some way to manipulate this to get what I need but I don't see it.
  
  Hints?
  Hint:
  $f(n+1) - f(n) = (10^{n+1} + 3*4^{n+3}+5) - (10^n + 3*4^{n+2}+5)$
  
  $\Rightarrow f(n+1) - f(n) = 9(10^n + 4^{n+2})$
  
  But $9|f(n)$ (by Induction Hypothesis)
  
  So............
  
  P.S: It should really be "9 divides f(n)", not vice versa
  
  3. Originally Posted by Isomorphism
  Hint:
  $f(n+1) - f(n) = (10^{n+1} + 3*4^{n+3}+5) - (10^n + 3*4^{n+2}+5)$
  
  $\Rightarrow f(n+1) - f(n) = 9(10^n + 4^{n+2})$
  
  But $9|f(n)$ (by Induction Hypothesis)
  
  So............
  
  P.S: It should really be "9 divides f(n)", not vice versa
  Fixed the mistake. Great idea. I can usually get the proof once I see it, but coming up with that by myself seems like an unlikely event. That hint was exactly what I needed. Thank you again.
explanation: The document contains a math question and a subsequent answer. The question is about proving that a function \( f(n) = 10^n + 3 \cdot 4^{n+2} + 5 \) is divisible by 9 for all non-negative integers \( n \). The answer provides a hint and a step-by-step approach to solving the problem using mathematical induction.
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distilabel_metadata.role: system
distilabel_metadata.content: 1. ## Induction with divisibility
  
  Define f: (N U {0}) -> N by $f(n) = 10^n + 3*4^{n+2}+5$ Prove that for all non-negative numbers, n, f(n) is divisible by 9.
  --------------------
  Ok so for this one P(n) is going to be the statement "f(n) is divisible by 9". P(0) holds and we can now assume that this holds true for all j<k.
  
  This part of induction always gets me, now I have to show P(j) -> P(k), or P(n) -> P(n+1).
  
  If 9 divides f(j), then there is some number, D, such that $9*D = 10^j + 3*4^{j+2}+5$. Now I know there's some way to manipulate this to get what I need but I don't see it.
  
  Hints?
  
  2. Originally Posted by Jameson
  Define f: (N U {0}) -> N by $f(n) = 10^n + 3*4^{n+2}+5$ Prove that for all non-negative numbers, n, f(n) divides 9.
  
  --------------------
  Ok so for this one P(n) is going to be the statement "f(n) divides 9". P(0) holds and we can now assume that this holds true for all j<k.
  
  This part of induction always gets me, now I have to show P(j) -> P(k), or P(n) -> P(n+1).
  
  If f(j) divides 9, then there is some number, D, such that $9*D = 10^j + 3*4^{j+2}+5$. Now I know there's some way to manipulate this to get what I need but I don't see it.
  
  Hints?
  Hint:
  $f(n+1) - f(n) = (10^{n+1} + 3*4^{n+3}+5) - (10^n + 3*4^{n+2}+5)$
  
  $\Rightarrow f(n+1) - f(n) = 9(10^n + 4^{n+2})$
  
  But $9|f(n)$ (by Induction Hypothesis)
  
  So............
  
  P.S: It should really be "9 divides f(n)", not vice versa
  
  3. Originally Posted by Isomorphism
  Hint:
  $f(n+1) - f(n) = (10^{n+1} + 3*4^{n+3}+5) - (10^n + 3*4^{n+2}+5)$
  
  $\Rightarrow f(n+1) - f(n) = 9(10^n + 4^{n+2})$
  
  But $9|f(n)$ (by Induction Hypothesis)
  
  So............
  
  P.S: It should really be "9 divides f(n)", not vice versa
  Fixed the mistake. Great idea. I can usually get the proof once I see it, but coming up with that by myself seems like an unlikely event. That hint was exactly what I needed. Thank you again.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a subsequent answer. The question is about proving that a function \( f(n) = 10^n + 3 \cdot 4^{n+2} + 5 \) is divisible by 9 for all non-negative integers \( n \). The answer provides a hint and a step-by-step approach to solving the problem using mathematical induction.
  
  Label: question-and-answer