Can you please explain to me why is (k(k+1) ) / 2 is an integer and I know it always even so can you help me prove that?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
Can you please explain to me why is (k(k+1) ) / 2 is an integer and I know it always even so can you help me prove that?
15)
n(n+5) is divisible by 6
for each
ノ
integer
Proof (ay induction)
P(O)= "o (o?+5) is divisible by 6"? 610 -0
nzo
Since
6.0= 0 V
P(K) > P(K+I): Let KE Z Ə
Assume
*0そオ
PCK) is true
"K (K?+5) is divisible by
that is,
So this mean that
6d - k(k²+5) for some
ivisibility
NTS: k+i(K+)+5) is divisible by 6
int gger
definition
d. by
of
(kt)((k1)*+5)= (kt1)(k*+2k +6)
= K3+3K?+ 8k t6
(k3+5k) +3K²+3k +6
k(k?+5) + 3k²+3K+6
Now,
by algebra
%3D
6d + 3k²+3K+6
by subsh tuton
= 6 (d + Klual) +) by algebra
2
integers closed
and products. Therefore ktl((K+1)*+5) is
divisibile by 6 by definition of divisibility.
Let e =
d + k(nt1)
le z since
2
undess sums
Transcribed Image Text:15) n(n+5) is divisible by 6 for each ノ integer Proof (ay induction) P(O)= "o (o?+5) is divisible by 6"? 610 -0 nzo Since 6.0= 0 V P(K) > P(K+I): Let KE Z Ə Assume *0そオ PCK) is true "K (K?+5) is divisible by that is, So this mean that 6d - k(k²+5) for some ivisibility NTS: k+i(K+)+5) is divisible by 6 int gger definition d. by of (kt)((k1)*+5)= (kt1)(k*+2k +6) = K3+3K?+ 8k t6 (k3+5k) +3K²+3k +6 k(k?+5) + 3k²+3K+6 Now, by algebra %3D 6d + 3k²+3K+6 by subsh tuton = 6 (d + Klual) +) by algebra 2 integers closed and products. Therefore ktl((K+1)*+5) is divisibile by 6 by definition of divisibility. Let e = d + k(nt1) le z since 2 undess sums
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
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,