Mathematics: Write out proof in paragraph form. include contradictions if necessary 5. prove that n(n+1) (2n+1) is always divisible by 6

Write out proof in paragraph form. include contradictions

sighgabbie

31.12.2021

If n=1 , then

$n(n+1)(2n+1)=1\cdot2\cdot3=6$

which is of course divisible by 6.

Assume the claim holds for n=k . Then if n=k+1 , we have

$(k+1)(k+2)(2k+3)=\underbrace{k(k+1)(2k+1)}+k(2k+3)+(k+2)(2k+3)+2k(k+1)$

(I use the distributive property of multiplication to extract the first term, which we've assumed is divisible by 6)

The claim holds for n=k+1 if

k(2k+3)+(k+2)(2k+3)+2k(k+1)

is also divisible by 6. With some manipulation we can express this as

(2k+2)(2k+3)+2k(k+1)

(k+1)(2(2k+3)+2k)

(k+1)(6k+6)

6(k+1)^2

which is clearly divisible by 6, so the claim is true for n=k+1 , and this completes the proof (by induction).

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ayoismeisalex

04.08.2022

One solution.

Step-by-step explanation:

So we have:

2x + 3 = 4x + 2

Simplifying gives;

4x - 2x = 3 - 2

2x = 1

x = 1/2

There is only one solution here.

*Note: This questions are simple. You just have to do a little more practice on algebra if you are stuck.

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