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B) Show that the following claim holds when for all n > 1 n (424) >...

Question

B) Show that the following claim holds when for all n > 1 n (424) >...

b) Show that the following claim holds when for all n > 1 n (424) > n(n+1)(n+2) i= 1

b) Show that the following claim holds when for all n > 1 n (424) > n(n+1)(n+2) i= 1
Computer Science Engineering
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Answers

for n=1

LHS= +i = 12 +1=1+1=2

RHS - 1(1+1)(1+2) 2/3)

so, LHS = RHS for n=1

lets assume LHS = RHS for n=k

k(k + 1)(k+2) so, (il +i)= i=1

for n=k+1

k+1 LHS = Σ12 +) i=1

= (+ 1)2 + (k + 1)) + Σ 1=1 ? + 1)

kk + 1)(k+2) = ((62 +2k + 1) + (k + 1)) + -

=(k^2+3k+2)+\frac{k(k+1)(k+2)}{3}

=(k^2+3k+2)+\frac{k(k^2+3k+2)}{3}

36² +9k +6+ (6 + 3.+ 25:)

=\frac{(k^3+6k^2+11k+6)}{3}

RHS=\frac{(k+1)(k+2)(k+3)}{3}

=\frac{(k+1)(k^2+5k+6)}{3}

=\frac{(k^3+5k^2+6k+k^2+5k+6)}{3}

=\frac{(k^3+6k^2+11k+6)}{3}

so,\;LHS=RHS\;for\;n=k+1\;given\;that\;LHS=RHS\;for\;n=k

so,\;\sum_{i=1}^n(i^2+i)=\frac{n(n+1)(n+2)}{3}\;is\;proved\;using\;induction

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