Question

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

Answer 1

Answer:

$a_n=-5n+106$

Step-by-step explanation:

Arithmetic sequences are linear.

Let's find the slope.

$\frac{a_{16}-a_{7}}{16-7}=\frac{26-71}{16-7}=\frac{-45}{9}=-5$.

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:

$a_n=-5n+106$.

Answer 2

Answer:

$a_{n} =101+(n-1)(-5)$

Step-by-step 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.

$a_{n} =101+(n-1)(-5)$