SOLUTION: bob made the following conjecture:if you add three consecutive integers,then their sum is equivalent to three times the secound integer. A.give 4 examples of 3 consecutive integ

Learn that:
"3 consecutive integers" means "3 whole numbers in a row"

One example of three whole numbers is a row is 5,6 and 7

Another example of 3 whole numbers in a row is 20, 21, and 22.

Another example of 3 whole numbers in a row is 1, 2, and 3.

Another example of 3 whole numbers in a row is 14,15, and 16.

In the first example above, 5, 6 and 7, 

the first one is 5, the second one is 6, and the third one is 7.

Now let's see what Bob said about those.

His first 6 words are:

"if you add three consecutive integers,..."

So let's stop and do that. We will add 5+6+7.  We get 18.

Let's see some more of what Bob said:

"then their sum..." 

"Their sum" means what we got when we added them, which was 18.

"is equivalent to..."

that means "is equal to"

"three times the second integer."

The second integer is 6.  So lets multiply 3 times 6. We get 18.

Now we ask?  Did we get 18 both times?  The answer is yes.

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In the second example above, 20, 21 and 22, 

the first one is 20, the second one is 21, and the third one is 22.

Now let's see what Bob said about those.

His first 6 words are:

"if you add three consecutive integers,..."

So let's stop and do that. We will add 20+21+22.  We get 63.

Let's see some more of what Bob said:

"then their sum..." 

"Their sum" means what we got when we added them, which was 63.

"is equivalent to..."

that means "is equal to"

"three times the second integer."

The second integer is 21.  So lets multiply 3 times 21. We get 63.

Now we ask:  Did we get 63 both times?  The answer is yes.

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In the third example above, 1, 2 and 3, 

the first one is 1, the second one is 2, and the third one is 3.

Now let's see what Bob said about those.

His first 6 words are:

"if you add three consecutive integers,..."

So let's stop and do that. We will add 1+2+3.  We get 6.

Let's see some more of what Bob said:

"then their sum..." 

"Their sum" means what we got when we added them, which was 6.

"is equivalent to..."

that means "is equal to"

"three times the second integer."

The second integer is 2.  So lets multiply 3 times 2. We get 6.

Now we ask:  Did we get 6 both times?  The answer is yes.

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In the fourth example above, 14, 15 and 16, 

the first one is 14, the second one is 15, and the third one is 16.

Now let's see what Bob said about those.

His first 6 words are:

"if you add three consecutive integers,..."

So let's stop and do that. We will add 14+15+16.  We get 45.

Let's see some more of what Bob said:

"then their sum..." 

"Their sum" means what we got when we added them, which was 45.

"is equivalent to..."

that means "is equal to"

"three times the second integer."

The second integer is 15.  So let's multiply 3 times 15. We get 45.

Now we ask:  Did we get 45 both times?  The answer is yes.

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We think his comnjecture is true because our answer was "yes" every time
in those four examples.

Let the three consecutive integers, whole numbers in a row

be N, N+!, and N+2

The first one is N, the second one is N+1, and the third one is N+2.

Now let's see what Bob said about those.

His first 6 words are:

"if you add three consecutive integers,..."

So let's stop and do that. We will add 

N + (N+1) + (N+2)

Removing the parentheses:

N + N + 1 + N + 2

We will combine like terms:

3N + 3 

We get 3N + 3.

Let's see some more of what Bob said:

"then their sum..." 

"Their sum" means what we got when we added them, which was 3N + 3.

"is equivalent to..."

that means "is equal to"

"three times the second integer."

The second integer is N+1.  So let's multiply 3 times N+1. 

[We have to put parentheses around (N+1)]

So to multiply that by 3 we have

3(N+1)

Now we use the distributive property to remove the parentheses:

3N + 3

Now we ask:  Did we get 3N + 3 both times?  The answer is yes.

Therefore we have proved that Bob is correct regardless of what N is.

Edwin