(10 pts) Prove the following by induction:
i=1n 3i – 2 = 3?2−?2
BASE CASE: Proving for n = 1
INDUCTIVE HYPOTHESIS: Assume that it works fine for n = k
INDUCTIVE CASE: Proving for n = k+1
HENCE PROVED
Induction proofs. a. Prove by induction: n sum i^3 = [n^2][(n+1)^2]/4 i=1 Note: sum is intended to be the summation symbol, and ^ means what follows is an exponent b. Prove by induction: n^2 - n is even for any n >= 1 10 points 6) Given: T(1) = 2 T(N) = T(N-1) + 3, N>1 What would the value of T(10) be? 7) For the problem above, is there a formula I could use that could directly calculate T(N)?...
DISCRETE MATHEMATICS Problem 3 (10 points) Use mathematical induction to prove the following statement for all n 21. For full credit, mention the base case (1pt), the induction hypothesis (1 pt) and the induction step (8 pts). 12 22 32
Combinatorics 10. (10 points) Page 101 #3 Prove by induction that for n 2 2 IT (1-)=F +1 2n
prove each of the following theorems using weak induction 1 Weak Induction Prove each of the following theorems using weak induction. Theorem 1. an = 10.4" is a closed form for an = 4an-1 with ao = 10. Theorem 2. an = (-3)"-1.15 is a closed form for an = -3an-1 with a1 = 15. Theorem 3. In E NU{0}, D, 21 = 2n+1 -1. Theorem 4. Vn e N, 2" <2n+1 - 2n-1 – 1. Theorem 5. In E...
9. Prove by mathematical induction: -, i = 1 + 2 + 3+...+ n = n(n+1) for all n > 2.
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose r is a real number other than 1. Prove using mathematical induction that for every nonnegative integer n, = 1-r^n+1/1-r. 3) Prove using mathematical induction that for every nonnegative integer n, 1 + i+i! = (n+1)!. 4) Prove using mathematical induction that for every integer n>4, n!>2^n. 5) Prove using mathematical induction that for every positive integer n, 7 + 5 + 3 +.......
Problem 44) Prove: n!> 2" for n24. Problem 45) Prove by induction: For n>0·AT- i=1
how do I prove this by assuming true for K and then proving for k+1 Use mathematical induction to prove that 2"-1< n! for all natural numbers n. Use mathematical induction to prove that 2"-1
Part I: Induction (90 pt.) (90 pt., 15 pt. each) Prove each of the following statements using induction, strong induction, or structural induction. For each statement, answer the following questions. a. (3 pt.) Complete the basis step of the proof. b. (3 pt.) What is the inductive hypothesis? c. (3 pt.) What do you need to show in the inductive step of the proof? d. (6 pt.) Complete the inductive step of the proof. 5. Let bo, bu, b2,... be...
prove by induction E N with n 2 2 we have 3 2 M-I E N with n 2 2 we have 3 2 M-I