Question

What is a21 of the arithmetic sequence for which a7=-19 and a10=-28

Answer 1

Answer:

-61

Step-by-step explanation:

So arithmetic sequences are linear.  This means no matter what two points you use from the sequence you will get the same slope (also known as common difference when talking about arithmetic sequences).

So we have two points given on this line:

(7,-19) and (10,-28).

Let's find the slope:

Line up the points and subtract vertically, then put 2nd difference over 1st difference.

Like this:

(  7  ,  -19)

-(10  ,  -28)

----------------

-3         9

So the slope is 9/-3 or -3 after reducing.

Now I'm going to use the following two points to find the 21st term:

(21,y) and (7,-19).

We are using these 2 points to find the slope. Line them up and subtract vertically, then put 2nd difference over 1st difference.

Like so:

 ( 21   ,  y)

- (   7  ,-19)

---------------

14        y+19

So the slope is (y+19)/14.

So since this is a line it shouldn't matter what two points you choose on it, the slopes should be the same.

So we have the following equation to solve:

$\frac{y+19}{14}=-3$

Multiply both sides by 14:

$y+19=-3(14)$

Simplify right hand side:

$y+19=-42$

Subtract 19 on both sides:

$y=-42-19$

Simplify right hand side:

$y=-61$

The 21st term is -61.