\( \require{color} \color{red} \ \ \text{ 0 is the first number for being true.} So, by the principle of mathematical induction P(n) is true for all natural numbers n. Use induction to prove that 10n + 3 × 4n+2 + 5, is divisible by 9, for all natural numbers n. P(1) ; 10 + 3 â‹… 64 + 5 = 207 = 9 â‹… 23. That is, 6k+4=5M, where M∈I. Induction proof - divisibility by 3. I need to prove that 7^n + 4^n +1 is divisible by 6 using induction, I habe gotten as far as the last step of n=k+1 which I am stuck on. Step 1: Show it is true for n=0. Now, we have to prove that P(k + 1) is true. \)That is, \( (k+2)(k+4) \) is divisible by 4.\( \begin{aligned} \displaystyle(k+2)(k+4) &= (k+2)k + (k+2)4 \\&= 4M + 4(k+2) \color{red} \ \ \text{ by assumption at Step 2} \\&= 4\big[M + (k+2)\big] \color{red} \text{, which is divisible by 4} \\\end{aligned} \)Therefore it is true for \( n=k+2 \) assuming that it is true for \( n=k \).Therefore \( n(n+2) \) is always divisible by \( 4 \) for any even numbers. Let us assume that P(n) is true for some natural number n = k. or K3 â€“ 7k + 3 = 3m, m∈ N         (i). Answer Save. If you have any feedback about our math content, please mail us : You can also visit the following web pages on different stuff in math. Step 2 : Let us assume that P(n) is true for some natural number n = k. \)\( 2(2+2) = 8\), which is divisible by 4.Therefore it is true for \(n=2\).Step 2:  Assume that it is true for \( n=k \).That is, \( k(k+2) = 4M \).Step 3:  Show it is true for \( n=k+2 \). 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Mathematical Induction: Divisibility This is part of the HSC Mathematics Extension 1 course under the topic Proof by Mathematical Induction. 2. Hot Network Questions It may not be in my best interest to ask a professor I have done research with for recommendation letters. Mathematical Induction question: Prove divisibility by $4$ of $5^n + 9^n + 2$ 4. mathematical induction for divisibility: Is this one a valid proof? Divisibility proofs by induction. Thus, P(k + 1) is true whenever P(k) is true. Induction proof, divisibility. If so why? 0. \( \require{color} \color{red} \ \ \text{ Odd numbers increase by 2.} Prove 6n+4 is divisible by 5 by mathematical induction. This isn't by induction, but I think it's a nice proof nonetheless, certainly more enlightening: $\displaystyle 5^n-1=(1+4)^n-1=\sum_ ... 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