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Help with arithmetic?

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What are the explicit equation and domain for an arithmetic sequence with a first term of 5 and a second term of 2?

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An arithmetic sequence has a common difference. Since the second term is 2 and the first term is 5, the common difference is 2 - 5 = -3 When you add the common difference to a term, you get the next term.
5 5 + (- 3) = 2 2 + (- 3) = -1 -1 + (- 3) = -4 -4 + (-3) = -7 The first 5 terms of the sequence are: 5, 3, -1, -4, -7
Since adding -3 is the same as subtracting 3, you can eliminate the first two choices because in those choices, you are subtracting multiples of 2. We need to subtract multiples of 3 to find subsequent terms. The answer has to be choice C or choice D.
Right! Wow this is making so much more sense than I thought it would
To figure out which one it is, look at choice C. \(a_n = 5 - 3(n - 1)\) for all integers n, such that \(n \ge 0\)
Use the equation of choice C., and plug in the first value of n. That would be n = 0. \(a_n = 5 - 3(n - 1)\) \(a_0 = 5 - 3(0 - 1)\) What value do you get for \(a_0\) ?
8!
Correct. According to choice C., the first term in the sequence is called \(a_0\), and it is 8. We were told the first term is 5, so choice C cannot be correct.
Wait. I got the two choices confused. I did choice D above, where n>= 0. Choice D. is eliminated.
Now let's look at choice C., which is C. \(a_n = 5 - 3(n - 1)\) for all integers n, such that \(n \ge 1\) \(a_n = 5 - 3(n - 1)\) \(a_1 = 5 - 3(1 - 1)\) Now for choice C., what do you get for the first term?
5
Correct. Choice C. gives us the correct first term. Now notice what happens as n becomes 2, then 3, then 4, etc. Each time you are subtracting 1 from the next integer, then multiplying it by -3. That means first you subtract 0 from 5 (what you did for term 1) Then you subtract 3 from 5, then you subtract 6 from 5, then you subtract 9 from 5, etc. giving you the terms of the sequence. The answer is C.
Okay, got it! Makes total sense. Thank you so SO much!
Is my answer right with this one? It's a similar problem. @mathstudent55
Let's see. Choice D. starts with n = 0. What do you get when you replace n with 0 below? \(a_n = 4(-12)^{n - 1} \) \(a_0 = 4(-12)^{0 - 1} \) What is \(a_0\) ?
-.333?
Oh sorry .333
\(a_0 = 4(-12)^{0 - 1} = 4 (12)^{-1} = \dfrac{4}{12} = \dfrac{1}{3}\) Correct. We are told the first term is 4. That means this cannot be the answer.
Yeah, I was hesitant on my answer. Glad I checked!
So it would be C., correct?
In the previous problem we had an arithmetic sequence. Notice this is a geometric sequence. Do you know the difference between a geometric sequence and an arithmetic sequence?
I don not.
In an arithmetic sequence (as we saw in the earlier problem) there is a common difference. If you add the common difference to a term, you get the next term. To find the common difference, subtract a term from the next term. In a geometric sequence there is a common ratio. If you multiply a term by the common ratio, you get the next term. To find the common ratio, divide a term by the previous term.
Look at the first term and the second term: 4 and -8. What is -8/4 = ?
Ooo, -2
So would it be B?
Right. The common ratio is -2, so you need an equation with a -2. That means we need A. or B.
The sequence follows by -8, 16, -32, 64 . . .
B. is correct because it gives us the correct first term, 4.
Okay, just asking, but why wouldn't it be A.?
Let's look at A. \(a_n =4(-2)^{n - 1} \); \(n \ge 0\) The first term uses n = 0: \(a_0 =4(-2)^{0 - 1} \) What is \(a_0\) using choice A.?
-2
Right, but we were told the first term is 4, so A. cannot be correct.
Ooo okay. Got it! Thank you so much!
I got 100%!! Thanks so much!

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