Given a sequence {a} such that a, > a > 0, and an+2 (an+1+an), for n=1,2,.......

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Given a sequence {a} such that a, > a₂ > 0, and an+2 (an+1+an), for n=1,2,.... Answer the following questions: (a) Show that for n 21, (-1)" 2n an+2-an= (a₁-a₂) and hence show that the sequence (a₁, a3, as,) is strictly decreasing and that the sequence (az, as, ae,) is strictly increasing. (b) For any positive integers m and n, show that ama2n-1. (c) Show that the two sequences {a,, as, as,...) and (a, a, a, converge to the same limit k, where 1 k=(a₁ +2a₂). 3 Given a sequence {a} such that a, > a₂ > 0, and an+2 (an+1+an), for n=1,2,.... Answer the following questions: (a) Show that for n 21, (-1)" 2n an+2-an= (a₁-a₂) and hence show that the sequence (a₁, a3, as,) is strictly decreasing and that the sequence (az, as, ae,) is strictly increasing. (b) For any positive integers m and n, show that ama2n-1. (c) Show that the two sequences {a,, as, as,...) and (a, a, a, converge to the same limit k, where 1 k=(a₁ +2a₂). 3

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a We have an2 a a1 a2a2 a a1 a2a1 a Thus we have an2 a a1 a2a1 a and hence an2 an1 an ... View the full answer