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Asked Oct 8, 2019
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Prove using induction: 

Exercise 3: Prove the following statements. I suggest induction.
(a) For all integers n > 0, 12 + 22 +...+n-n(n+1) (2n + 1)
6
(b) For all n > 0, (1+2+.+ n)2 = 13 + 23 + .+n3.
(c) For n
2, n < (2 -n1)".
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Exercise 3: Prove the following statements. I suggest induction. (a) For all integers n > 0, 12 + 22 +...+n-n(n+1) (2n + 1) 6 (b) For all n > 0, (1+2+.+ n)2 = 13 + 23 + .+n3. (c) For n 2, n < (2 -n1)".

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Expert Answer

Step 1

We’ll answer the first question since the exact one wasn’t specified. Please submit a new question specifying the one you’d like answered.

We first check for n=1 

We got 1=1, which is true

So for n=1 the statement is true.

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1(1+1)(2(1)+1) 12 6 (2)(2+1) 1= 6 (2)(3) 6

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Step 2

Let the statement is true for some m>0.

So we assume P(m) is true. 

 

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P(m): 12+22+32+....+m?= m(m+1)(2m+1) 6

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Step 3

We have to show that P(m+1) is true when...

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P(m+1): 12+22+3?+....+m2+(m-+1?= (m+1)((m+1)+1)(2(m+1)+1) 6 Or P(m+1) 12+22+32+...+m2+(m+1?-m+1)(m+2)(2m+3) We have to check if P(m+1) is true.

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