Question

If you are good at arithmetic sequences in math please help I am giving more points

Answer 1

`a_(n) =101+(n-1)(-5)`

Explanation:

Let's start by finding the common difference of this arithmetic sequence.

The change from
`a_(7)` to
`a_(16)` is -45 because it decreased 45.

`a_(16)` is 9 terms away from
`a_(7)`.

If we divide the change by the difference in n, then we can find the difference between each term, or the common difference.

-45/9 = -5, so the common difference is -5.

Now, we want to find
`a_(1)`.

We can do this using the common difference. Subtracting the common difference from
`a_(7)` six times should tell us what
`a_(1)` is, so let's do that.

`a_(1)` = 71 - (-5) - (-5) - (-5) - (-5) - (-5) - (-5)

A reminder, subtracting a negative is basically the same as adding it.

`a_(1)` = 71 + 5 + 5 + 5 + 5 + 5 + 5

`a_(1)` = 101

With the common difference and
`a_(1)`, we now have all the parts we need to write a rule for the nth term of this sequence.

The formula for the nth term of a sequence is

`a_(n) =a+(n-1)d`, with
`a_(n)` being the nth term, a being
`a_(1)`, and d being the common difference.

We can just substitute the common difference and a to get our formula.

`a_(n) =101+(n-1)(-5)`

Answer 2

`a_n=-5n+106`

Explanation:

Arithmetic sequences are linear.

Let's find the slope.

`(a_(16)-a_(7))/(16-7)=(26-71)/(16-7)=(-45)/(9)=-5`.

So the slope is -5.

We know this about our equation:

`a_n=-5n+b`

We need to find
`b`.

Let's use a point on the line like
`(7,71)`.

Input the point into our equation:

`71=-5(7)+b`

`71=-35+b`

Add 35 on both sides:

`106=b`

The equation for this line/arithmetic sequence is:

`a_n=-5n+106`.