Suppose we are given a sequence of numbers x1, x2, x3, …,xn,…
Have the following properties: x1 = 2 and xn+1 = 2n/(n+1) xn for all n Є N
Use Mathematical induction to prove that xn = (2^n)/n for all n Є N
Thanks in advance
Hello, lisaak,
I assume that you know mathematical induction.
Step 1: n = 1 so $\displaystyle x_1=\frac{2^1}{1}=\frac{2\cdot 1}{1}=2$ is true.
step 2: Assume that $\displaystyle x_{n}=\frac{2^n}{n}$ is true.
step 3: Induction
$\displaystyle x_{n+1}=\frac{2n}{n+1}\cdot \frac{2^n}{n}$. Rearrange:
$\displaystyle x_{n+1}=\frac{n}{n}\cdot \frac{2 \cdot 2^n}{n+1}=\frac{2^{n+1}}{n+1}$
EB