Finding the Sum of Geometric Sequences with Given Constraints

[wvcaudill2]

Homework Statement

The sum of the first six terms in a geometric sequence of real numbers is 252.
Find the sum of the first four terms when the sum of the first two terms is 12.

Homework Equations

S_n = A₁ - A₁Rⁿ
divided by 1 - R

R $$\neq 1$$

(I can't figured out how to make the "divided by" sign)

The Attempt at a Solution

Well, I subtracted 12 from 252 to find the sum of the last 4 terms and got 240. I could think of nothing more to do after this though.

 

[rock.freak667]

If the sum of the first 6 number is 252 and the sum of the first two is 12. What does that say about the common ratio R? (more than 1 or less than 1?)

When you get that, then just apply your formula for S_n (Note that the formula you posted is valid for |R|<1)

 

[wvcaudill2]

Okay, so the common ratio must be more than 1 since the numbers are getting bigger.

If my formula only works for |R|< 1, then how can I use that formula?

 

[rock.freak667]

Okay, so the common ratio must be more than 1 since the numbers are getting bigger.

If my formula only works for |R|< 1, then how can I use that formula?

For |R|>1, the formula changes slightly


$$S_n = \frac{A(R^n-1)}{R-1}$$

Now S₆ = 252 and S₂= 12

 

[wvcaudill2]

Okay, so do I just solve for R in one equation and then plug it into the other? I still have A though, which I assume I also need to find.

I am still unsure about how to go about solving this problem.

 

[rock.freak667]

Okay, so do I just solve for R in one equation and then plug it into the other? I still have A though, which I assume I also need to find.

I am still unsure about how to go about solving this problem.

Well before you start solving, what are your two equations in A and R?

 

[rasmhop]

When you get that, then just apply your formula for S_n (Note that the formula you posted is valid for |R|<1)

This is incorrect. His formula is correct for all R not equal to 1. If you look at his formula and yours you will see that you just give him the exact same formula he posted himself (except you factored out A_1).

The condition |R|<1 is only required for infinite geometric series (i.e. when you let n go to infinity).

The rest of rock.freak667's advice is solid.

BTW the following manipulation may help you (the exact same information can be extracted from your formula):
$$S_2 = A+AR$$
$$S_6 = (A+AR) + (AR^2 + AR^3) + (AR^4 + AR^5) = (A+AR) + R^2(A+AR) + R^4(A+AR) = (1+R^2+R^4)(A+AR)$$

 

[wvcaudill2]

Im not quite sure what you mean. But, when I reworked the first equation for R, I got:

R = 6th root of 252(R-1)+1 divided by A

and

R = square root of 12(R-1)+1 divided by A

I don't think this is the right direction though.

 

[rasmhop]

You want to find R without referring to A so try dividing S_6 by S_2. You should get:
$$\frac{S_6}{S_2} = \frac{1-R^6}{1-R^2} = R^4 + R^2 + 1$$
which let's you solve for R^2.

 

[wvcaudill2]

How did you get either of those equations for S₆ or S₂?

Earlier, you stated
S₆ = (1 + R² + R⁴)(A + AR)

and

S₂ = (A + AR)


now, if you take S₆/S₂ would that not equal


(1 + R² + R⁴)(A + AR)/(A + AR) = (1 + R² + R⁴)

?

If so, then when solved for R I get R = the sixth root of 20, or about 1.6475. Is this correct?

 

[rasmhop]

How did you get either of those equations for S₆ or S₂?

Factoring out the common factors from your formula we get:
$$S_n = \frac{A}{1-R}(1-R^n)$$
So in general:
$$\frac{S_n}{S_m} = \frac{ \frac{A}{1-R}(1-R^n)}{ \frac{A}{1-R}(1-R^m)} = \frac{1-R^n}{1-R^m}$$

Earlier, you stated
S₆ = (1 + R² + R⁴)(A + AR)

and

S₂ = (A + AR)now, if you take S₆/S₂ would that not equal (1 + R² + R⁴)(A + AR)/(A + AR) = (1 + R² + R⁴)

?

Yes.

If so, then when solved for R I get R = the sixth root of 20, or about 1.6475. Is this correct?

No. S_6/S_2 = 21 so you get:
$$R^4 + R^2 +1 = 21$$
or written differently:
$$R^4 + R^2 - 20 = 0$$
Which has the solution (the other solution is negative):
$$R^2 = \frac{-1+\sqrt{1+80}}{2} = \frac{-1+9}{2} = 4$$
Now you can find A from one of your equations and finally evaluate S_4.

 

[wvcaudill2]

How did you get to R² and its solution?

 

[rasmhop]

How did you get to R² and its solution?

The quadratic formula:

If $x^2 + bx + c = 0$, then x is one of the following:
$$x_- = \frac{-b-\sqrt{b^2-4c}}{2} \qquad x_+ = \frac{-b+\sqrt{b^2-4c}}{2}$$
Have you not seen this? (I just assumed you knew it) If you haven't we may be able to find an alternative way to solve this.

 

[wvcaudill2]

I thought I recognized the Quadratic formula, I just didnt think it could be used here.

I can solve for S₄ now. Thank you for your help!