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# proof by induction format

Show this by factoring as $2k(k+1)(2k+1)$. The argument can be shaped to fit weak induction structure (what you started to do), and then one has to do the cases-trick for 4k^3+6k^2+2k. Thanks for contributing an answer to Mathematics Stack Exchange! I've been trying to work on this problem and I can't seem to solve it. MathJax reference. However, we can show that n = k-5 implies that the statement is true for k+1, so we need to expand the base case to include everything up to n = 6. Prove by induction that: $$f(x) = e^x \sin(x)$$ $$f^{(n)}(x)=2^{\frac{n}{2}}e^x\sin(x+\frac{n\pi}{4})$$ I Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. However, for most other values of l we would end up with a final expression where it is hard to prove that the expression is divisible by 12. Weak induction is no help (writing n as a product of primes does nothing for doing the same to n+1); strong induction works very well, it kicks in in the case that n+1 is not a prime already. n = 5:  12|(54 – 52) = 12|(625- 25) = 600 = 5012 Show this by factoring as $2k(k+1)(2k+1)$. So our property P P is: n3 + 2n n 3 + 2 n is divisible by 3 3. That is, the statement “if blablabla is true for all natural m