1. Prove that Σ(Φ) = 2(2)" - 1) i=0 for any nonnegative integer n. Complete the basis step of the proof. What is the inductive hypothesis? What do you need to show in the inductive step of the proof? Complete the inductive step of the proof.
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- Show that StartFraction d Over dx EndFraction (ln kx)equalsStartFraction d Over dx EndFraction ln x, given that xgreater than 0 and kgreater than 0 is a real number.6. Prove that if m and n are integers and mn is even, then m is even or n is even.Suppose that the equation ax b .mod n/ is solvable (that is, d j b, whered D gcd.a; n/) and that x0 is any solution to this equation. Then, this equation has exactly d distinct solutions, modulo n, given by xi D x0 C i.n=d / fori D 0; 1; : : : ; d 1
- Prove that for any integer n, n6k-1 is divisible by 7 if gcd(n,7)=1 and k is a positive integer.(i) Prove by cases for any given integer n,the number (n3-n) is even. (Ii)Prove by contradiction there are no integers x and y such that x2=4y+2.Let the sequence (n) be recursively defined by x1 = √2 and Xn+1 = √√2+xn, n≥ 1. Show that (n) converges and evaluate its limit.