Question

Help with arithmetic?

Answer 1

@uri @SolomonZelman @mathstudent55

Answer 2

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

Answer 3

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.

Answer 4

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

Answer 5

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.

Answer 6

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

Answer 7

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

Answer 8

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 \(a_0\) ?

Answer 9

8!

Answer 10

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.

Answer 11

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

Answer 12

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?

Answer 13

5

Answer 14

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.

Answer 15

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

Answer 16

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

Answer 17

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 \(a_0\) ?

Answer 18

-.333?

Answer 19

Oh sorry .333

Answer 20

\(a_0 = 4(-12)^{0 - 1} = 4 (12)^{-1} = \dfrac{4}{12} = \dfrac{1}{3}\)

Correct.

We are told the first term is 4. That means this cannot be the answer.

Answer 21

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

Answer 22

So it would be C., correct?

Answer 23

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?

Answer 24

I don not.

Answer 25

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.

Answer 26

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

Answer 27

Ooo, -2

Answer 28

So would it be B?

Answer 29

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

Answer 30

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

Answer 31

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

Answer 32

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

Answer 33

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.?

Answer 34

-2

Answer 35

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

Answer 36

Ooo okay. Got it! Thank you so much!

Answer 37

I got 100%!! Thanks so much!