Help with arithmetic?

- anonymous

Help with arithmetic?

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- anonymous

@uri @SolomonZelman @mathstudent55

- anonymous

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

- anonymous

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- mathstudent55

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.

- mathstudent55

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

- mathstudent55

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.

- anonymous

Right! Wow this is making so much more sense than I thought it would

- mathstudent55

To figure out which one it is, look at choice C.
\(a_n = 5 - 3(n - 1)\) for all integers n, such that \(n \ge 0\)

- mathstudent55

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 \(a_0\) ?

- anonymous

8!

- mathstudent55

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.

- mathstudent55

Wait. I got the two choices confused. I did choice D above, where n>= 0.
Choice D. is eliminated.

- mathstudent55

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?

- anonymous

5

- mathstudent55

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.

- anonymous

Okay, got it! Makes total sense. Thank you so SO much!

- anonymous

Is my answer right with this one? It's a similar problem. @mathstudent55

##### 1 Attachment

- 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 \(a_0\) ?

- anonymous

-.333?

- anonymous

Oh sorry .333

- mathstudent55

\(a_0 = 4(-12)^{0 - 1} = 4 (12)^{-1} = \dfrac{4}{12} = \dfrac{1}{3}\)
Correct.
We are told the first term is 4. That means this cannot be the answer.

- anonymous

Yeah, I was hesitant on my answer. Glad I checked!

- anonymous

So it would be C., correct?

- mathstudent55

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?

- anonymous

I don not.

- mathstudent55

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.

- mathstudent55

Look at the first term and the second term: 4 and -8.
What is -8/4 = ?

- anonymous

Ooo, -2

- anonymous

So would it be B?

- mathstudent55

Right. The common ratio is -2, so you need an equation with a -2.
That means we need A. or B.

- anonymous

The sequence follows by -8, 16, -32, 64 . . .

- mathstudent55

B. is correct because it gives us the correct first term, 4.

- anonymous

Okay, just asking, but why wouldn't it be A.?

- mathstudent55

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.?

- anonymous

-2

- mathstudent55

Right, but we were told the first term is 4, so A. cannot be correct.

- anonymous

Ooo okay. Got it! Thank you so much!

- anonymous

I got 100%!! Thanks so much!

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