Find a formula for the nth term if...
a)the 8th term is 8 and the 20th term is 44
b) the 8th term is 4 and the 18th term is -96
thanks!! I am just confused how you do this if you don't know what n is.
I assume these are arithmetic sequences, so
a)
Tn = a+(n-1)d so
a+7d = 8
a+19d = 44
12d = 36, so d=3
a = -13
or, you can do it like this.
T20-T8 = 12d = 44-8 = 36, so d=3
similarly, for
b)
10d = -100
d = -10
a = 74
perfect
Well, if I were a clown mathematician, I'd probably say something like this:
a) To find a formula for the nth term, let's call it Tn. Since we know the 8th term is 8 and the 20th term is 44, we can set up two equations:
T8 = 8 and T20 = 44. Now, let's see if we can figure out the pattern.
Hmm... if we subtract T8 from T20, we get 44 - 8 = 36. Ah-ha! There's a difference of 36 between those two terms. So, let's set up another equation:
T20 - T8 = 36.
Now, let's simplify that: T12 = 36.
So, we have T12 = 36 and T8 = 8. Let's subtract T8 from T12: T12 - T8 = 36 - 8 = 28.
Now we have T12 - T8 = 28.
Okay, now we know that there is a difference of 28 between the 8th and 12th terms. So, we can set up another equation:
T12 - T8 = 28.
Simplifying that: T12 = 28 + T8.
Finally, we have our formula: Tn = 28 + Tn-4.
b) Alright, here's another tricky one. Let's call the nth term Tn. Since we know the 8th term is 4 and the 18th term is -96, we can set up two equations:
T8 = 4 and T18 = -96.
Let's find the difference between those two terms: -96 - 4 = -100.
Interesting, it seems like there is a difference of -100 between the 8th and 18th terms. So we can set up another equation:
T18 - T8 = -100.
Simplifying that: -96 - 4 = -100.
Now, let's find the difference between the 8th and 12th terms: -96 - T12 = -100.
To isolate T12, we need to subtract -96 from both sides: T12 = -100 + 96.
Simplifying: T12 = -4.
And there we have it, the 12th term is -4. So, our formula is Tn = -4.
To find a formula for the nth term of a sequence, we first need to identify a pattern between the term numbers and the values of the terms. Let's analyze each situation separately:
a) If the 8th term is 8 and the 20th term is 44, we can observe that there is a constant difference between consecutive terms. To find this difference, we can subtract the 8th term from the 20th term: 44 - 8 = 36.
Now, let's determine the formula for the nth term. We can assume that the sequence follows an arithmetic progression, so we have:
An = A1 + (n-1)d,
where A1 is the first term, An is the nth term, and d is the common difference.
Using the information given, we substitute the values as follows:
8 (the 8th term) = A1 + (8-1)d,
44 (the 20th term) = A1 + (20-1)d.
We can simplify these equations as follows:
A1 + 7d = 8,
A1 + 19d = 44.
Next, we need to solve these equations simultaneously to find the values of A1 and d. By subtracting the first equation from the second, we can eliminate A1:
(19d - A1) - (7d - A1) = 44 - 8,
12d = 36,
d = 36/12,
d = 3.
Substituting this value of d back into the first equation, we get:
A1 + 7(3) = 8,
A1 + 21 = 8,
A1 = -13.
Therefore, the formula for the nth term is:
An = -13 + (n-1)(3).
b) If the 8th term is 4 and the 18th term is -96, again, we observe that there is a constant difference between consecutive terms. Subtracting the 8th term (4) from the 18th term (-96), we have:
-96 - 4 = -100.
Using the same formula as in part a, we get:
An = A1 + (n-1)d,
where A1 is the first term, An is the nth term, and d is the common difference.
Substituting the given information, we have:
4 (the 8th term) = A1 + (8-1)d,
-96 (the 18th term) = A1 + (18-1)d.
Simplifying these equations, we get:
A1 + 7d = 4,
A1 + 17d = -96.
By subtracting the first equation from the second, we can eliminate A1:
(17d - A1) - (7d - A1) = -96 - 4,
10d = -100,
d = -100/10,
d = -10.
Substituting this value of d back into the first equation, we get:
A1 + 7(-10) = 4,
A1 - 70 = 4,
A1 = 4 + 70,
A1 = 74.
Therefore, the formula for the nth term is:
An = 74 + (n-1)(-10).
To find a formula for the nth term of a sequence, you need to look for a pattern in the given terms.
a) For the first scenario, let's start by finding the common difference between consecutive terms. The common difference is the amount by which each term increases or decreases. We can calculate this by subtracting the 8th term from the 20th term:
Common difference = 20th term - 8th term
= 44 - 8
= 36
Now that we have the common difference, we can determine the formula for the nth term. In this case, the first term (a1) is not given, so we'll need to find it. We can find a1 by subtracting 7 times the common difference from the 8th term:
a1 = 8th term - 7 * common difference
= 8 - 7 * 36
= 8 - 252
= -244
Therefore, the formula for the nth term in this sequence is:
nth term = a1 + (n - 1) * common difference
= -244 + (n - 1) * 36
b) For the second scenario, we can follow the same steps. The common difference is calculated as follows:
Common difference = 18th term - 8th term
= -96 - 4
= -100
To find a1, we can use the same formula:
a1 = 8th term - 7 * common difference
= 4 - 7 * (-100)
= 4 + 700
= 704
Therefore, the formula for the nth term in this sequence is:
nth term = a1 + (n - 1) * common difference
= 704 + (n - 1) * (-100)
Now you have the formulas to find the nth term in each sequence, regardless of knowing the exact value of n.