Can you help me explain why (k(k+1)) / 2 is an integers 15)n(n+5) is divisible by 6for eachノintegerProof (ay induction)P(O)= "o (o?+5) is divisible by 6"? 610 -0nzoSince6.0= 0 VP(K) > P(K+I): Let KE Z ƏAssume*0そオPCK) is true"K (K?+5) is divisible bythat is,So this mean that6d - k(k²+5) for someivisibilityNTS: k+i(K+)+5) is divisible by 6int ggerdefinitiond. byof(kt)((k1)*+5)= (kt1)(k*+2k +6)= K3+3K?+ 8k t6(k3+5k) +3K²+3k +6k(k?+5) + 3k²+3K+6Now,by algebra%3D6d + 3k²+3K+6by subsh tuton= 6 (d + Klual) +) by algebra2integers closedand products. Therefore ktl((K+1)*+5) isdivisibile by 6 by definition of divisibility.Let e =d + k(nt1)le z since2undess sums

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Can you help me explain why (k(k+1)) / 2 is an integers

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 46E: Use generalized induction and Exercise 43 to prove that n22n for all integers n5. (In connection...

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Question

Can you help me explain why (k(k+1)) / 2 is an integers

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

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