MAT102 Term Test 3 Review

• 3
  • $u_{n+1}=\frac{u_n^2}{u_{n-1}}$
  • $u_0=1,u_1=2$
  • $u_n=2^n$
  • Claim: $u_n$ can be represented as $u_n=2^n$
  • Base Case:
    • $u_0=2^0=1$
    • $u_1=2^1=2$
    • $u_2=2^2=4$
    • True
  • Induction Hypothesis:
    • Assume for some arbitrary $n\in \mathbb{N}$ that $u_n=2^n$ is true
  • Induction Step:
    • We want to show that $u_n=2^n⟹u_{n+1}=2^{n+1}$
    • $u_{n+1}=\frac{u_n^2}{u_{n-1}}$
    • $u_{n+1}=\frac{2^{2n}}{u_{n-1}}$
    • $u_{n+1}=\frac{2^{2n}}{2^{n-1}}$
    • $u_{n+1}=2^{2n}{2}^{-(n-1)}$
    • $u_{n+1}=2^{2n-(n-1)}$
    • $u_{n+1}=2^{2n-n+1}$
    • $u_{n+1}=2^{n+1}$