Yet ty = 1 and tn-1 = 1 - mutts tr for n Z 1a Show lim tn existsb What do you think lim tn is Use induction to showr tn-號号。d Repeat part b

(a) We can rewrite tn as tn = (1 - mutts)^n : tr. As tn-1 = (1 - mutts)^(n-1) : tr, we can divide tn by tn-1 to get:

tn/tn-1 = ((1 - mutts)^n : tr) / ((1 - mutts)^(n-1) : tr) tn/tn-1 = (1 - mutts) : tr

Since mutts > 0 and tr > 0, we know that 0 < (1 - mutts) : tr < 1. Therefore, tn/tn-1 is a bounded sequence, and by the ratio test, we know that lim tn exists.

(b) We can see that tn = (1 - mutts)^n : tr is a decreasing geometric sequence. As n approaches infinity, the sequence approaches 0, so we think that lim tn = 0.

(c) Base case: t1 = (1 - mutts)^1 : tr = (1 - mutts) : tr = tn-1, so the statement is true for n = 1.

Induction step: Assume that tn-1 = (1 - mutts)^(n-1) : tr. Then:

tn = (1 - mutts)^n : tr tn = (1 - mutts) * (1 - mutts)^(n-1) : tr tn = (1 - mutts) * tn-1

Therefore, tn = (1 - mutts)^n : tr is true for all n Z 1.

(d) Using the formula from part (c), we can see that:

lim tn = lim [(1 - mutts) * tn-1] lim tn = (1 - mutts) * lim tn-1

Since we think that lim tn-1 = 0, we can substitute to get:

lim tn = (1 - mutts) * 0 lim tn = 0

Therefore, we confirm that lim tn = 0