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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?

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
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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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
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