The sum of the sixth and ninth terms of an arithmetic sequence is 20 and the product of the sixth and ninth terms is 64. Find the tenth term in the sequence if the first term in the sequence is negative.

Let denote the th term in the sequence. Then

and so on down to

where is the common difference between terms in the sequence.

This means we have

Given that and , we can substitute the relationships above to solve for :

From the first equation, we get

Substitute this into the second equation and solve for :

from which it follows that the first term is (because we're told it's negative).

Then the common difference is , and so the 10th term in the sequence is

2: 10

step-by-step explanation:

delaney would like to make a 5lb nut mixture that is 60% peanuts and 40% almonds.

she has several pounds of peanuts and several pounds of a mixture that is 20% peanuts and 80% almonds.

let 'p' represent the number of pounds of peanuts needed to make the new mixture, and  

let 'm' represent the number of pounds of the 80% almond-20% peanut mixture.

:

p + m = 5

therefore

m = (5-p)

write an equation based on the percent of peanuts

1p + .2m = .6(5)

replace m with (5-p)

p + .2(5-p) = .6(5)

p + 1 - .2p = 3

p - .2p = 3 - 1

.8p = 2

p = 2/.8

p = 2.5 lbs of peanuts required

then

5 - 2.5 = 2.5 lbs of the peanut/almond mixture required

:

a) what is the system that models this situation?

p + m = 5

1p + .2m = .6(5)