Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
7. (+1) (n+3 (n+3)(n+2) 2 for n ≥ -1

7. (+1) (n+3 (n+3)(n+2) 2 for n ≥ -1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Can you please help with #7 and #9? TIA

Justify the equations in 6-9 either by deriving them from for- mulas in Example 9.7.1 or by direct computation from Theo- rem 9.5.1. Assume m, n, k, and r are integers. 6. (m+k₁) - = m + k, for m+k≥ 1 1 n+3 7. (+³) = (n+3)(n+2) 2 , for n ≥ -1 n+1 8. (7) = 1, for k-r≥0 k r 2n 9. (²2) for n ≥ 0 n
Transcribed Image Text:Justify the equations in 6-9 either by deriving them from for- mulas in Example 9.7.1 or by direct computation from Theo- rem 9.5.1. Assume m, n, k, and r are integers. 6. (m+k₁) - = m + k, for m+k≥ 1 1 n+3 7. (+³) = (n+3)(n+2) 2 , for n ≥ -1 n+1 8. (7) = 1, for k-r≥0 k r 2n 9. (²2) for n ≥ 0 n
Theorem 9.7.1 Pascal's Formula Let n and r be positive integers and suppose r ≤n. Then n+ (" + ¹) = ( ² ) + ( ) · r Proof (algebraic version): Let n and r be positive integers with r ≤n. By Theorem 9.5.1, n! ( ²₁ ) + ( )= + (r − 1)!(n − (r − 1))! - r!(n-r)! n! n! + (r − 1)!(n − r + 1)!r!(n-r)!* To add these fractions, a common denominator is needed, so multiply the numera- tor and denominator of the left-hand fraction by r and multiply the numerator and denominator of the right-hand fraction by (n-r+1). Then n! r n! (²₁) + ( ) (r − 1)!(n −r + 1)! r (n-r+1) r!(n-r)! (nr+1) n-n! -n!-r+n! n!.r (n-r+1)!r(r 1)! (nr+1)(n-r)!r! _n!+r+n!•n__n!r+n! = n!(n + 1) (n+1-r)!r! (n-r+1)!r! (n + 1)! = ((n + 1) -r)!r! = = ("+¹).
Transcribed Image Text:Theorem 9.7.1 Pascal's Formula Let n and r be positive integers and suppose r ≤n. Then n+ (" + ¹) = ( ² ) + ( ) · r Proof (algebraic version): Let n and r be positive integers with r ≤n. By Theorem 9.5.1, n! ( ²₁ ) + ( )= + (r − 1)!(n − (r − 1))! - r!(n-r)! n! n! + (r − 1)!(n − r + 1)!r!(n-r)!* To add these fractions, a common denominator is needed, so multiply the numera- tor and denominator of the left-hand fraction by r and multiply the numerator and denominator of the right-hand fraction by (n-r+1). Then n! r n! (²₁) + ( ) (r − 1)!(n −r + 1)! r (n-r+1) r!(n-r)! (nr+1) n-n! -n!-r+n! n!.r (n-r+1)!r(r 1)! (nr+1)(n-r)!r! _n!+r+n!•n__n!r+n! = n!(n + 1) (n+1-r)!r! (n-r+1)!r! (n + 1)! = ((n + 1) -r)!r! = = ("+¹).
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question

Thank you. Can you please explain why you multiplied by 2! in the denominator in step 2? How does that 2! get there?

Solution
Bartleby Expert
SEE SOLUTION
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS