SOLUTION: Let a + ar + ar^2 + ar^3 + ... be an infinite geometric series. The sum of the series is 3. The sum of the cubes of all the terms is 5. Find the common ratio.

Question 1209387: Let
a + ar + ar^2 + ar^3 + ...
be an infinite geometric series. The sum of the series is 3. The sum of the cubes of all the terms is 5. Find the common ratio.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The given series is

a+ar+ar^2+ar^3+...

The sum of the series is 3:

[1]

The series consisting of the cubes of the terms of the given series is

a^3r^3+a^3r^6+a^3r^9+...

The sum of that series is 5:

[2]

To find the common ratio r, solve [1] for a in terms of r and substitute in [2].

By inspection, r=1 is one solution to that equation. However r=1 produces an infinite geometric series that has no sum.

Use synthetic division to remove the root x=1 to find the other two roots.


  1 | 22 -81  81 -22
    |     22 -59  22
    +---------------
      22 -59  22   0

The remaining quadratic is

Use the quadratic formula to find that the other two solutions are

There are two possible values for the common ratio r:

ANSWERS:

RELATED QUESTIONS
Let a + ar + ar^2 + ar^3 + \dotsb be an infinite geometric series. The sum of the... (answered by CPhill,greenestamps)

Can someone help me with this? Thank you. infinite geometric series a+ar+ar^2+ar^3+...... (answered by rothauserc)

the second term of the infinite geometric series, a + ar + ar^2 + ar^3...... is... (answered by radikrr,stanbon)

Question: The sum of three numbers in a GP is 14. If the first two terms are each... (answered by ikleyn)

what is the sum of a+ar+ar^2.....to infinity when... (answered by ikleyn)

Please help me solve this equation: Given the sum of a and a3 = 3 and the sum of a4... (answered by lynnlo)

Please help..What does it mean "the sum of the terms involving the odd powers of r... (answered by ikleyn)

{{{S=a+ar+ar^2+ar^3+ar^(n-1)}}}, what happens to the sum when... (answered by satyareddy)

The sum of first three terms of a finite geometric series is -(7/10) and their product is (answered by stanbon)