2. (i) Recall the Pascal recurrence m + m k (^) + (^ + ¹). k 1 = Prove by induction on n that if k is a nonnegative integer, and n ≥k is an integer, then +...+ m + k n+ 1 ( ₁² ) + 6( n + ¹) = ₁ 3 n+ (2) = (x + 1). This is the diagonal property of Pascal's triangle, a.k.a. a hockey stick iden- tity. (ii) Show that = n³ for n ≥ 1.
2. (i) Recall the Pascal recurrence m + m k (^) + (^ + ¹). k 1 = Prove by induction on n that if k is a nonnegative integer, and n ≥k is an integer, then +...+ m + k n+ 1 ( ₁² ) + 6( n + ¹) = ₁ 3 n+ (2) = (x + 1). This is the diagonal property of Pascal's triangle, a.k.a. a hockey stick iden- tity. (ii) Show that = n³ for n ≥ 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![2. (i) Recall the Pascal recurrence
m
(₁)+()-("+¹).
1
m
1³ +2³+...+n³
Prove by induction on n that if k is a nonnegative integer, and n ≥k is an
integer, then
(*) + (* + ¹).
(iii) By (ii), we can write
n+ 1
( ₁ ) + 6( " + ¹) =
3
+...+
This is the diagonal property of Pascal's triangle, a.k.a. a hockey stick iden-
tity.
(ii) Show that
Explain how this leads to
m
1³ +2³+
k
n+
(2) = (x + 1).
=
(1) + ( ² ) +---+ (1) + 6 [(3) + ( 3 ) +----+ (^ ;+ ¹)]
3
·+n³.
= n³ for n ≥ 1.
n² (n + 1)²
4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7a498f9-6dcf-4975-8966-bdc41b8fc68e%2F819cdd8b-55c6-4019-b3ad-26cea8160bec%2Fzko7o3_processed.png&w=3840&q=75)
Transcribed Image Text:2. (i) Recall the Pascal recurrence
m
(₁)+()-("+¹).
1
m
1³ +2³+...+n³
Prove by induction on n that if k is a nonnegative integer, and n ≥k is an
integer, then
(*) + (* + ¹).
(iii) By (ii), we can write
n+ 1
( ₁ ) + 6( " + ¹) =
3
+...+
This is the diagonal property of Pascal's triangle, a.k.a. a hockey stick iden-
tity.
(ii) Show that
Explain how this leads to
m
1³ +2³+
k
n+
(2) = (x + 1).
=
(1) + ( ² ) +---+ (1) + 6 [(3) + ( 3 ) +----+ (^ ;+ ¹)]
3
·+n³.
= n³ for n ≥ 1.
n² (n + 1)²
4
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