The sequence {bn}∞n=1 is defined as b1 = 5 , b2= 13, bn = 5 bn −1 − 6 bn −2 for n ≥ 3. Prove bn = 2n + 3n for all n ≥ 1.

Let a and b be positive numbers with a>b. Let a1 be their arithmetic mean and b1 their geometric mean. a1 = a+b/2      b1 = √ab Repeat this process so that, in general, an+1= an+bn/2     bn+1= √ab (a) Use mathematical induction to show that an>an+a>bn+1>bn (b) Deduce that both {an} and {bn} are convergent. (c) show that limn-->infinity an = limn-->infinity bn. Guass call the commone value of thes limits the arithmetic-geometric mean of the numvers a and b.

Compute b4​ b) recursively.

|(-1)*+2 2n – 5 2 4. Determine if converges. (n+1)² + 5