(r+1)2" = 3"-(2n +3) %3D r30
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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Question
![## Exercise 4
For each of the following, give two proofs: one algebraic and one using counting.
### (a)
\[
\sum_{r=0}^{n} \binom{n}{r} (r + 1) 2^r = 3^{n-1}(2n + 3)
\]
### (b)
\[
\sum_{a_1 + a_2 + a_3 + a_4 = n} \binom{n}{a_1, a_2, a_3, a_4} a_1 a_2 3^{a_3} = n(n-1)6^{n-2}
\]
(Here \( n \) is fixed, and the sum is over all 4-tuples \( a_1, a_2, a_3, a_4 \) of nonnegative integers whose sum is \( n \).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdb89a50-aa64-4601-b71c-fab5c9074739%2F80b79e24-a2cc-4b95-9d92-932f5693ac91%2Fbyg15lj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Exercise 4
For each of the following, give two proofs: one algebraic and one using counting.
### (a)
\[
\sum_{r=0}^{n} \binom{n}{r} (r + 1) 2^r = 3^{n-1}(2n + 3)
\]
### (b)
\[
\sum_{a_1 + a_2 + a_3 + a_4 = n} \binom{n}{a_1, a_2, a_3, a_4} a_1 a_2 3^{a_3} = n(n-1)6^{n-2}
\]
(Here \( n \) is fixed, and the sum is over all 4-tuples \( a_1, a_2, a_3, a_4 \) of nonnegative integers whose sum is \( n \).)
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