How do I prove by induction,
If t1 = 1, tn+1 = 1 + Ö(1 + tn) prove a) tn < 3 for all n Î N
b) tn+1 > tn for all n Î N
Hello, polymerase!
I'll help you with part (a) . . .
If $\displaystyle t_1 = 1,\;t_{n+1} = 1 + \sqrt{1 + t_n}$, prove by induction:
a) $\displaystyle t_n < 3$ for all $\displaystyle n \in N$
Verify $\displaystyle S(1)$. . Is $\displaystyle t_1 < 3?$
. . $\displaystyle t_1 = 1 < 3$ . . . yes!
Assume $\displaystyle S(k)$ is true: .$\displaystyle t_k \:< \:3$
Add 1 to both sides: .$\displaystyle 1 + t_k\:<\:4$
Take the square root of both sides: .$\displaystyle \sqrt{1 + t_k} \:<\:2$
Add 1 to both sides: .$\displaystyle \underbrace{1 + \sqrt{1 + t_k}}_{\text{This is }t_{k+1}} \:<\:3$
Therefore: .$\displaystyle t_{k+1}\:<\:3$
. . The inductive proof is complete.