0 (A) STATE THE BINOMIAL THEOREM AND use it TO DETERMINE THE COEFFICIENT OF xs in...
Powers of 11 2. a. Use the Binomial Theorem on powers of (10+1) to demonstrate that the numbers b. At what point does the pattern not continue? Why does the pattern no longer c. Compute 11 x 11 using the standard algorithm for multiplication. Do likewise d. Explain how the standard algorithm for multiplication of powers of 11 relates to in the first few rows of Pascal's Triangle resemble the digits of the powers of 11 work? with 121 x...
Question 3* For any n,T EN the biomial coefficient ( is the coefficient of in the expansion of (1 + z)". (E.g. (4) 6 because (1 + z)4-1 + 4x + 612 + 4r' + re) In particular, 0 whenever r >n and ()) for all nEN*. These facts, together with Pascal's identity (")+ )(), facilitate the calculation of the value of () for any particular values of n and r via the well-know 'Pascal's triangle'. a) Use Pascal's identity...
12. Use the binomial theorem to find the coefficient of xayh in the expansion of (5x2 +2y3)6, where a) a 6, b-9 b) a 2, b 15. c) a 3, b 12. d) a 12, b 0 e) a 8, b 9
The Arithmetical Triangle sparalleles arta Blaise Pascal's 1955 work Treatise on the Arithmetical Triangle contains a collection several results already known about the "Pascal's Triangle" for more than 500 years, as well as applications of the triangle to probability. Among these results are the Hockey-Stick Theorem the fact that the sum of a row is a power of 2, as well as the following fact about ratios: の(k+1) = (k + 1):(n-k) We have actually already seen this fact somewhere...
For the following linear system: 2x--3x +5x2-7x4-0 X-2x+3x-2xs-0 3x3-3 x-x-3x: 0 X:+8 x 9 x3+11 xs-0 (1) Represent this linear system in the form Ax = b. (ii) Explain what the null space of the coefficient matrix A is in terms of the linear system (ii) Find a basis for the null space of A. (iv) Find the rank and nullity of matrix A.
please hell with 13, 14, 15, 16, 17 13. Use the binomial theorem to expand: (w-11) 14. Use mathematical induction to prove the following: 3+ 7 + 11 + 15 + ... + (4n-1) = n(2n+1) 15. Write an expression for the n" term of the sequence 2! 3! 4! 5! 6! 7! 4'5'6'7'8'9' 16. The sixth term of an arithmetic sequence is 20.6, and the ninth term is 30.2. a. Find the nth term b. Find the 20 term...
Main topic and problems for the final project The main purpose of the project is to introduce you how to use a in an computer as a research tool Introductory Discrete Mathematics. In this project you will be asked to show how the Fibonacci sequence {Fn} is related to Pascal's triangle using the following identities by hand for small and then by computers with large n. Finally, to rove the identity by mathematical arguments, such as inductions or combinatorics. I...
Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Rolling a single "loaded" die 8 times, keeping track of the numbers that are rolled. A. Procedure results in a binomial distribution. B. Not binomial: there are more than two outcomes for each trial. C. Not binomial: the trials are not independent. D. Not binomial: there are too many trials.
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
Determine whether you can use the normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use the binomial distribution to find the indicated probabilities. A survey of adults in a region found that 52% have encountered fraudulent charges on their credit cards. You randomly select 100 adults in the region. Complete parts (a) through (d) below (a) Find the probability...