Transcribed TextTranscribed Text

1. (2 X 4=8 pts) Find the shortest possible closed formula equal to each. X - Allowed key stroke: binomial coefficient : \binom(x)(y) y a. n 1 - n 2 + n 3 - + (-1)n+1 - n n b. 0 n 1 + 2 n ( 2n + ) n Hint: Write it down with the sigma notation. Use two binomial coefficient identities learned in class. Which are they? n 2 n 2 n 2 n 2 c. 1 + 2 + 3 + + n 1 2 3 n Hint: Use three learned binomial coefficient identities. d. (") - 1 ) 1 n - m n - 1 ) m - n 1 n j - ) ( - j - 1 m - 1 ) 2. (5pts) Below it is discussed how to simplify k-1 kxk (x 1) with change of index. Fill in the five blanks with the shortest possible expressions to complete the sentences. Allowed keystroke: power 'N' Put the sum in S. Consider the two sums in n n S - X S = kxk - k (a) k=1 k=1 In the second sigma notation, change the index k into j=k+1. So we have n (c) (1 S = - (d) k=1 Then re-write j back to k. Find that it is equal to n (1 - x)S = k=1 Xk - (e) xn-1 The sigma notation in the right hand side is X as we remember in class. Solve this for S to x-1 find the closed formula. 3. (7pts) Fill in the five blanks to complete the argument below to approximate n! as a double inequality. (a), (b), (c): 1pt, (d), (e): 2pts Allowed key strokes: natural logarithm base'e', In '\ln' Let t = Inn! where In X means the natural logarithm of X. Observe n t = (a) k=1 Noting f (x) = In x is monotonically increasing, find with the telescope method that t > (b) That is, use the fact that the area of the red boxes is greater than the area between f(x) and x=0. f(c) = In x x 1 2 3 4 n Then shift the red boxes to the right by distance 1, removing the last one. Now the area is smaller than that between f(x) and x=0, so t<__(c). As a result, (d) < n! < (e) achieving our goal.

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

#   - Write the answer in each line never changing the labels such as 1-a.

1-a. 1
1-b. \binom(2n + 1)(n + 1)
1-c. n \binom(2n - 1)(n - 1)
1-d. 1

2-a. x^(k + 1)
2-b. 2
2-c. n + 1
2-d. (j - 1)x^j
2-e. nx^(n + 1) - x^(n + 1)

3-a. \ln k
3-b. 0
3-c. \ln (n - 1)
3-d. 1
3-e. (n - 1)^(n - 1)...

By purchasing this solution you'll be able to access the following files:
Solution.pdf and

for this solution

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

Find A Tutor

View available Discrete Math Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.

Upload a file
Continue without uploading

We couldn't find that subject.
Please select the best match from the list below.

We'll send you an email right away. If it's not in your inbox, check your spam folder.

  • 1
  • 2
  • 3
Live Chats