Question

The sum of the first three terms of a finite geometric series is7/6 and their product 1/27 is .

(Hint: Use , a, and ar to represent the first three terms, respectively.)
The three numbers are _______, ________, and _________.

Answer 1

1/6, 1/3, 2/3

Explanation:

Given data:

The sum of the first three terms of a finite geometric series is7/6 and their product 1/27 is .

Let a/r , a and ar be the three terms of a finite geometric series then:

a/r + a + ar = 7/6

and

(a/r) x (a) x (ar) = 1/27

Now first solving for a:

solving second equation

a^3r/r = 1/27

a^3= 1/27

a = 1/
`\sqrt[3]{27}`

a=1/3

Now solving for r:

Solving first equation

a/r + a + ar = 7/6

Putting value of a= 3 in above equation

1/3r + 1/3 + r/3 = 7/6

(1+r+r^2)/3r= 7/6

6(1+r+r^2)= 7(3r)

6+ 6r+ 6r^2= 21r

6r^2 - 15r +6=0

r=2

Hence the first three terms of a finite geometric series are

a/r= (1/3)/(2)

= 1/6

a= 1/3

ar= 1/3 (2)

=2/3 !