00:01
So the first part of this problem, actually i'll label this number one, is you're told that the 100th term in an arithmetic sequence is 77, and we know our common difference, which we refer to as the letter d is equal to three.
00:11
We want to know the first three terms.
00:13
Well, remember, an arithmetic sequence is a sequence in which to get from one term to the next, you add a constant.
00:19
Now, could we continue to subtract three from 77 until we get down to the first three terms? we could, but that might take a while.
00:25
But we have a formula to find any term in the sequence.
00:28
The formula is a -sub -n, is equal to a sub 1 plus d times the quantity of m minus 1.
00:36
Well, in this case, what we need to find is a sub 1.
00:39
That's our first term.
00:40
N represents the term in the sequence.
00:43
D is our common difference.
00:44
And a sub n is the nth term in the sequence.
00:47
So because we know that the 100th term is 77, we know that 77 would be a sub ed.
00:52
That means n would equal to 100.
00:55
And we already know d is equal to 3.
00:57
So we can use this equation to find a sub 1.
01:00
So now let's go ahead and simplify.
01:02
What would have 100 minus 1, which is 99, and 3 times 99 is 297.
01:08
So we have 77 equal to a sub 1 plus 297.
01:13
So to solve, we just subtract 297 from both sides of our equation, and 77 minus 297 is equal to negative 220, which means that a sub 1 is negative 220.
01:25
So that would be our first term in the sequence.
01:27
Well, to find the second term, remember our common difference is three.
01:32
So we're going to add three to negative 220, which is equal to negative 217.
01:37
So to find the third term, we take our second term negative 217 and we add our common difference, which would be negative 214.
01:45
So now we've found the first three terms in this arithmetic sequence.
01:49
All right.
01:50
So now for the second part of the problem, number two, we're told that we want to know which term n of the arithmetic.
01:58
Sequence, 6, 9, 12, dot, dot, dot is 90.
02:02
So in other words, we're going to take that same formula from before, a sub n is equal to a sub 1 plus d times n minus 1.
02:10
And we're looking for n.
02:12
Well, we know a sub 1, that's our first term, which is 6.
02:15
Our common difference, to get from one term to the next, we're adding 3, so it actually happens to be 3 again.
02:21
We don't know n, so they're going to leave that in here, but we know the nth term in the sequence is 90, so we can substitute that in for a sub -n.
02:29
So now we just have to solve the equation.
02:31
So first, i'm going to subtract six from both sides.
02:34
So we'll have 90 minus 6, which is 84, equal to 3 times n minus 1.
02:39
Then what i can actually do is divide both sides of our equation by 3, because those 3s will cancel out, and 84 divided by 3 is equal to 28.
02:48
So 28 is equal to n -1.
02:51
So to solve for n, we simply add one to both sides, and 28 plus 1 is 29.
02:56
So that means 90 is the 29th term in the sequence.
03:01
All right.
03:02
And then for the last question here, so let's call this number three, we're told that we have an arithmetic sequence where the first term is 5, so a sub 1 is 5...