# Math Help - mathematical induction recursion

1. ## mathematical induction recursion

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

2. Hello, polymerase!

If $t_1 = 1,\;t_{n+1} = 1 + \sqrt{1 + t_n}$, prove by induction:

a) $t_n < 3$ for all $n \in N$

Verify $S(1)$. . Is $t_1 < 3?$
. . $t_1 = 1 < 3$ . . . yes!

Assume $S(k)$ is true: . $t_k \:< \:3$

Add 1 to both sides: . $1 + t_k\:<\:4$

Take the square root of both sides: . $\sqrt{1 + t_k} \:<\:2$

Add 1 to both sides: . $\underbrace{1 + \sqrt{1 + t_k}}_{\text{This is }t_{k+1}} \:<\:3$

Therefore: . $t_{k+1}\:<\:3$
. . The inductive proof is complete.