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# Arithmetic sequence problem

Video transcript

We are asked, what is the value
of the 100th term in this sequence? And the first term is 15, then
9, then 3, then negative 3. So let's write it like
this, in a table. So if we have the term, just
so we have things straight, and then we have the value. and then we have the
value of the term. I'll do a nice little
table here. So our first term
we saw is 15. Our second term is 9. Our third term is 3. I'm just really copying this
down, but I'm making sure we associate it with
the right term. And then our fourth term
is negative 3. And they want us to figure out
what the 100th term of this sequence is going to be. So let's see what's happening
here, if we can discern some type of pattern. So when we went from the
first term to the second term, what happened? 15 to 9. Looks like we went down by 6. It's always good to think
about just how much the numbers changed by. That's always the simplest
type of pattern. So we went down by 6,
we subtracted 6. Then to go from 9 to 3, well,
we subtracted 6 again. And then to go from 3 to
negative 3, well, we subtracted 6 again. So it looks like, every
term, you subtract 6. So the second term is going to
be 6 less than the first term. The third term is going to be
12 from the first term, or negative 6 subtracted twice. So in the third term, you
subtract negative 6 twice. In the fourth term,
you subtract negative 6 three times. So whatever term you're looking
at, you subtract negative 6 one less than
that many times. Let me write this down just so--
Notice when your first term, you have 15, and
you don't subtract negative 6 at all. Or you could say you subtract
negative 6 0 times. So you can say this is 15 minus
negative 6 times-- or let me write it better
this way --minus 0 times negative 6. That's what that first
term is right there. What's the second term? This is 15. We just subtracted negative
6 once, or you could say, minus 1 times 6. Or you could say plus
1 times negative 6. Either way, we're subtracting
the 6 once. Now what's happening here? This is 15 minus 2 times
negative 6-- or, sorry --minus 2 times 6. We're subtracting a 6 twice. What's the fourth term? This is 15 minus-- We're
subtracting the 6 three times from the 15, so minus
3 times 6. So, if you see the pattern here,
when we have our fourth term, we have the term
minus 1 right there. The fourth term, we have a 3. The third term, we have a 2. The second term, we have a 1. So if we had the nth term, if we
just had the nth term here, what's this going to be? It's going to be 15 minus--
You see it's going to be n minus 1 right here. Right? When n is 4, n minus 1 is 3. When n is 3, n minus 1 is 2. When n is 2, n minus 1 is 1. When n is 1, n minus 1 is 0. So we're going to have this term
right here is n minus 1. So minus n minus 1 times 6. So if you want to figure out
the 100th term of this sequence, I didn't even have
to write it in this general term, you can just look
at this pattern. It's going to be-- and I'll do
it in pink --the 100th term in our sequence-- I'll continue
our table down --is going to be what? It's going to be 15 minus 100
minus 1, which is 99, times 6. right? I just follow the pattern. 1, you had a 0 here. 2, you had a 1 here. 3, you had a 2 here. 100, you're going to
have a 99 here. So let's just calculate
what this is. What's 99 times 6? So 99 times 6-- Actually you
can do this in your head. You could say that's going to
be 6 less than 100 times 6, which is 600, and
6 less is 594. But if you didn't want to do
it that way, you just do it the old-fashioned way. 6 times 9 is 54. Carry the 5. 9 times 6, or 6 times 9 is 54. 54 plus 5 is 594. So this right here is 594. And then to figure out what 15--
So we want to figure out what 15 minus 594 is. And this can sometimes be
confusing, but the way I always process this in my head
is, I say that this is the exact same thing as the negative
of 594 minus 15. And if you don't believe
me, distribute out this negative sign. Negative 1 times 594
is negative 594. Negative 1 times negative
15 is positive 15. So these two statements
are equivalent. This is much easier for my
brain to understand. So what's 594 minus 15? We can do this in our heads. 594 minus 14 would be 580,
and then 580 minus 1 more would be 579. So that right there is 579, and
then we have this negative sign sitting out there. So the 100th term in our
sequence will be negative 579.