Question

Find the next term in this arithmetic sequence: 2/3, 2, 3 1/3, ____

Find the eleventh term in this arithmetic sequence: .5, .7, . . .

Find the tenth term in this arithmetic sequence: 27, 24, . . .

Answer 1

Problem 1:
`4(2)/(3)`

Problem 2: 2.5

Problem 3: 0

Explanation:

If it tells you it is arithmetic, it is give you a huge hint. Arithmetic sequences have common differences. That means if you do:

`\text{term}-\text{ term right before}` you will get the same number.

Problem 1:

`\text{Term}-{Previous}`

`2-(2)/(3)`

`1(1)/(3)`

So the common difference is
`1(1)/(3)`.

The next term can be found by adding
`1(1)/(3)` to the previous.

`3(1)/(3)+1(1)/(3)=4(2)/(3)`.

Problem 2:

0.5 , 0.7 , ....

Start with finding the common difference:

`0.7-0.5`

`0.2` is the common difference.

The common difference can be used to find the next term given it's previous term.

So you have:

First term is 0.5 .

Second term is 0.7 .

Third term is 0.7+ 0.2=0.9 .

Fourth term is 0.9 + 0.2=1.1 .

You could find a pattern so you don't have to find all the terms before the 11th.

So let's start over:

First term is 0.5 .

Second term is 0.5+0.2 .

Third term is 0.5+0.2(2).

Fourth term is 0.5+0.2(3).

.............

nth term is 0.5+0.2(n-1).

The thing in parenthesis was always one less than term number we wanted to find.

The 11th term is given by 0.5+0.2(10).

Let's simplify:

0.5+0.2(10)

0.5+2

2.5

2.5 is the 11th term.

Problem 3:

We are going to do something similar that we did for the problem before this one.

First common difference is given by:

Term-Previous

24-27

-3

The common difference is -3. (It is going down by 3 each time.)

The tenth will be given by 27+-3(9).

Let's simplify:

27+-3(9)

27+-27

0