I can't figure this out. Please help! I will give points for best answer.
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z = 5n + 1
First, note that this is a problem about arithmetic sequences. The way that you would go about figuring out this problem (or the way that I did it, anyhow), is to start by pair off each n value with the z values side by side:
1 6
2 11
3 16
4 21
5 26
By looking at the consecutive z values, we should see that the k'th z value is just the (k - 1)'th z value plus 5. For example, the 3rd z value (i.e. when n = 3) is 16, so the 4th z value is 16 + 5 = 21, and the 5th z value is 21 + 5 = 26.
Now that we know the relationship between the terms (sometimes called the "common difference"), we can try to come up with a formula to derive the n'th term just by being given the value of n (so that we don't have to count up by 5's every time). In general, the way we do this is to multiply n by the common difference, then add some kind of corrective term if we need it. In other words,
z = dn + c
where d is the common difference (in this case, we know already that it is 5), and c is the corrective term. So how do you figure out what the corrective term is? Well, you have a bunch of trial values for n and z already, so you can just set it up like an algebra problem using one of those known pairs, then solve for c. I will pick the first one for simplicity:
6 = 5(1) + c
6 = 5 + c
1 = c
we could have just as readily picked the 4th one (or any of the others):
21 = 5(4) * c
21 = 20 + c
1 = c
and we will get the same answer every time. So, since we now know the value of c, we have our formula:
z = 5n + 1
Furthermore, if you have to do more of these arithmetic sequence problems, you can do them by looking for the values of c and d, then just using z = dn + c
Good luck!
First, note that this is a problem about arithmetic sequences. The way that you would go about figuring out this problem (or the way that I did it, anyhow), is to start by pair off each n value with the z values side by side:
1 6
2 11
3 16
4 21
5 26
By looking at the consecutive z values, we should see that the k'th z value is just the (k - 1)'th z value plus 5. For example, the 3rd z value (i.e. when n = 3) is 16, so the 4th z value is 16 + 5 = 21, and the 5th z value is 21 + 5 = 26.
Now that we know the relationship between the terms (sometimes called the "common difference"), we can try to come up with a formula to derive the n'th term just by being given the value of n (so that we don't have to count up by 5's every time). In general, the way we do this is to multiply n by the common difference, then add some kind of corrective term if we need it. In other words,
z = dn + c
where d is the common difference (in this case, we know already that it is 5), and c is the corrective term. So how do you figure out what the corrective term is? Well, you have a bunch of trial values for n and z already, so you can just set it up like an algebra problem using one of those known pairs, then solve for c. I will pick the first one for simplicity:
6 = 5(1) + c
6 = 5 + c
1 = c
we could have just as readily picked the 4th one (or any of the others):
21 = 5(4) * c
21 = 20 + c
1 = c
and we will get the same answer every time. So, since we now know the value of c, we have our formula:
z = 5n + 1
Furthermore, if you have to do more of these arithmetic sequence problems, you can do them by looking for the values of c and d, then just using z = dn + c
Good luck!
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z is also known as the nth term
Here we have the first term a = 6
Common difference d = 5
z = a + (n-1)*d
z = 6 + (n-1)*5
z = 6 + 5n - 5
z = 5n + 1
Here we have the first term a = 6
Common difference d = 5
z = a + (n-1)*d
z = 6 + (n-1)*5
z = 6 + 5n - 5
z = 5n + 1
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n=(z-1)/5
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z=5n+1