In a geometric progression of positive terms, the 5th term is 9 times the 3rd term and the sum of the 6th and 7th terms is 972. find the
a) common ratio
b) sum of the first 6 terms

a) 3

b) 364

Step-by-step explanation:

A geometric sequence in explicit form is where is the first term and is the common ratio.

We are given:

.

What is a) r?

What is b) the sum of the first 6 terms?

So I'm going to use my first equation and use my explicit form to find those terms in terms of r:

Divide both sides by :

.

So part a is 3.

Now for part b).

We want to find .

So far we have:

.

We also haven't used:

.

I'm going to find these terms in terms of r (r=3).

You have like terms to add:

Divide both sides by 972:

The first term is 1 and the common ratio is 3.

The terms we wrote can be simplify using a substitution for the first term as 1:

.

Now we just need to find the sum of those six terms:

1+3+9+27+81+243=364.

region b will make the inequalities y < 2x + 1 and y > -x - 2 true ⇒ answer a

step-by-step explanation:

* lets study the graph and the inequality to solve the question

- the inequality y > -x - 2

∵ the form of the inequality is y > mx + c , m is the slope of the line

  and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = -2 that means the line intersect the y-axis at point (0 , -2)

∵ the sign of the inequality is >

∴ we shad the part over the line

  region a and b

- the inequality y < 2x + 1

∵ the form of the inequality is y > mx + c , m is the slope of the line

  and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = 1 that means the line intersect the y-axis at point (0 , 1)

∵ the sign of the inequality is <

∴ we shad the part under the line

  region b and c

∵ the common region between the two lines is region b

∴ region b will make the inequalities y < 2x + 1 and y > -x - 2 true