Prove the following statements using weak induction: (a) E1 fi = fn+2 – 1, where f is the Fibonacci Sequence: f1 = f2 = 1, fn = fn-1+ fn-2.

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.

12n – 5n is divisible by 7, for all n  0. Give the formal proof of one of the following (your choice), using Mathematical Induction.

Part D Please

1) Guess a formula for the sum 1-3+5-7+.. +(-1)*-(2n – 1) =>(-1)-(2i – 1) 1=1 and prove your formula is correct using induction