Consecutive Integer Problem - 1

>> I'm going to cover consecutive integer problems
so let's start off with examples of two consecutive integers.
Consecutive means just one after the next
so if the first one is 5, the next number would be 6,
those are two numbers in a row right?
How about let's say 10 and 11,
those are 2 consecutive integers.
You could also have a negative number so like you could have -8
and the next number after -8 would be -7,
think of a number line etc. So these are some examples
of 2 consecutive integers so let's say instead
of a specific integer that you know you're gonna have
to use variables.
You don't know what the first one is
but let's say the first one is N what would be the next integer?
So we don't know what N stands for what would be the next one?
Well see what I did on each of these
if I knew what the first one is to get
to the second one you add one
so if the first number is N the second number would be M+1.
So to define two consecutive integers you would say the first
one is N and then you would say the second one would be N+1,
that's how you could define 2 consecutive integers.
What if you were asked to define 3 consecutive integers?
Well again let's just think of some examples first.
So examples might be 5, 6, and 7 right,
or how about 22, 23, and 24.
Or let's say we have some negative numbers like -10,
then the next one would be -9 and -8
so hopefully you're getting the idea
so if I were gonna define 3 consecutive integers I'm gonna
say the first number, let's call it this way, the first,
the second, and the third number.
Alright we always want to start off with the first one
and what would the next consecutive integer be?
Add 1 but how would I get the third number?
You add one again so that's how you would define 3
consecutive integers.
I'm gonna do a problem were we have
to define consecutive integers and then we'll go onto what
if it's talking about consecutive odd integers,
or consecutive even integers.
We'll start off with just consecutive integer problem.
Find 3 consecutive integers whose sum is negative 81.
Alright so we know we're looking for 3 consecutive integers
so that's how we want to start off defining our variables
so we've got the first number let's remember to call that N
and since they're consecutive it's easy M+1 is the next one
and the third number is M+2 not it says their sum is -81
so basically we're adding the first number
to the second number, to the third number,
and we should get -81.
I get the idea I'm gonna end
up having some negative numbers here
so that's how we're gonna do our equation
so the first number is just N plus the second number is N+1,
I'm gonna use a different color so it's easier to see it,
and the third number is M+2
so those are our 3 consecutive integers and that equals -81
so there's our equation.
Now we're going to go ahead and solve this equation
so we're gonna add the like terms on the left side,
3N+3 is -81, subtract 3 from both sides and be careful
with your positive and negative signs here.
We have 3N is -84 and then we need to divide both sides
by 3 so that N equals -28.
Alright now the question is find 3 consecutive integers so now
that we know what N is that stands for the first one
so the first number must be -28
and then what would the next one be?
Add one to that and add one to that, now let's add those up.
They're all gonna be negative so I could just put a negative sign
and if I add that up I get 81 which is right.
Not used to adding negatives up and down you might want
to show your check this way.
-28+-27+-26= - 81 ok that's the check and now of course we have
to answer the question that's being asked.
Find 3 consecutive integers so we say the integers are -28,
-27, and -26 and that's an example
of the consecutive integer problem.
Alright here's another consecutive integer problem.
Find 2 consecutive integers
if 3 times the smaller integer is 12 more
than twice the larger integer.
Alright so we have to begin by defining our variables
and we've got a first and second but I'm gonna write that instead
as the smaller and the larger number this time,
since it happens to use the word smaller and larger,
but these are consecutive integers
so if the first one's N,
the second one will be 1 more which is M+1.
Alright so what's it say?
3 times the smaller is 12 more than twice the larger
so we can do a little bit of English
and algebra together here.
What it's saying here I'm gonna do 3 times whatever
that smaller number is right?
That's gonna be the same is 12 more.
That means I'm gonna add 12 right to something, to what?
Twice the larger, twice the larger number alright
so let's do that.
3 times the smaller will be 3 and the smaller one is N
so I'm gonna put in N for the smaller, that's the same
as 2 times the larger number.
Now the larger number is N+1 so you need to put
that in parenthesis N+1 right and then add 12
and there's our equation.
Alright so let's go ahead and solve it, 3N is 2N+2+12
and I'm gonna subtract 2N from both sides
and then I'm also going to also add the 2 and 12
on the right side of the equal side so I get N =14
so I know what N is and what's N stand for?
The small, the first number so M+1,
the next number is gonna be 15 right?
So it looks like the numbers are 14 and 15
but let's make sure this all checks out.
Alright so it says 3 times the smaller number,
3 times 14 is 42 ok.
Is that the same thing?
Is 12 more than twice the larger?
Well if you got 15 what's twice 15?
That means 2 times 15, 30 if I add 12 to that I also get 42.
So you could check to see that makes sense that 14
and 15 are the two numbers we're looking for.
Yeah and so that will be the answer here.
The numbers are 14 and 15
[ pause in speaking ]
that's how to do a consecutive integer problem.
Now I'm not gonna have time to do a problem
about consecutive even and odd integers
but I'm gonna do a little introduction here.
Let's start out with some examples
of 3 consecutive even integers.
Alright you know like 4, 6, and 8,
those are 3 consecutive even integers or how about 14,
16 and 18 and of course you might have some negative numbers
like -20, -18 and -16 so if you look
at the first integer how would you get to the next integer?
You add 2 so if the first integer is N what would the next
one be?
M+2 and what would the one after that be?
So this is 2 consecutive integers.
If I wanted 3 the next one would be M+4.
So instead of adding 1 we're gonna go by 2s.
Now let's look at examples of 3 consecutive odd integers.
Alright let's do some odd integers.
How about 7, 9 and 11, or 23, 25, 27,
etc. Well what happens here?
If you know what the first one is let's say its N how would you
get to the next odd integer?
You don't' want to add one because you'd be
at an even integer so you have to add 2
to get another odd integer and then to get
to the next odd integer you'd have to add 2 again.
So check this out whether you're talking about even
or odd integers they're both defined as N and M+2 and M+4
if you're doing it for 3 in a row and we'll work on those
in another video, problems with those.