How Can Arithmetic and Geometric Progressions Reveal Values of a and b?

[matadorqk]

Arithmetic/Geometric Progression Problem SOLVED

Homework Statement

The Three terms a, 1, b are in arithmetic progression. The three terms 1, a, b are in geometric progression. Find the value of a and of b given that a cannot equal b.

Homework Equations

Arithmetic:
Here, Un is the nth term.U1 is the first term.
Un=U1+(n-1)d
Here, Sn is the Sum of n terms.
Sn=n/2(2U1+(n-1)d)
Sn=n/2(U1+Un)
Geometric
Un=U1 x r^n-1
Below, r can't equal 1.
Sn=U1(r^n-1)/r-1
Sn=U1(1-r^n)/1-r
Below, |r|<1
S=U1/1-r

The Attempt at a Solution

Well, let's establish a couple basic things.
Arithmetic
U1=a
U2=1
U3=b
U2-U1=d
U3-U2=d

U2-U1=U3-U2

So:
1-a=b-1
a+b=2
a=2-b
b=2-a
Then, using the formulas:
Un=U1 + (n-1)d
Sn=n/2(2U1 + (n-1)(d))=n/2(U1+Un)

We can find that:
a1+a2+a3=s3
Which is:
a1+a1+d+a1+2d
Which is:
3a+3d=s3

This is what I have so far, Ill update if I get more.. I am kind of stuck on this and another problem, school starts in 5 hours..(which assumes I won't sleep) so help asap would be appreciated..Thanks!

Now, I went on to the geometric side, where logic, one of my true strongs, hit me with a reasonable answer.
If r=u2/u1 and r=u3/u2
a/1=b/a
a=b/a
The only possible and logical thing that popped in my head was, wel A has to be negative.
So, I plot in -1, but then notice, in arithmetic sequences, b would end up as 1 aswell, which is obviously impossible for an arith.s.
Therefore, -2 seemed fit. I plot it, everything seems fine, my common difference would be 3, so my next term should be 4.
Lets plot that into a couple formulas.
a=b/a
-2=4/-2. YES
a+b=2
-2+4=2 YES

I revised every one, it does work. Sometimes, logic works better for me than anything, I guess that's my way of solving problems.

SOLVED

 

[cristo]

There is, of course, a way to do it without guessing an answer. From the arithmetic series, you obtain the equation b=2-a, and from the geometric, a=b/a, or, b=a^2. Hence, eliminating b yields a^2+a-2=0, which can be solved to give (a+2)(a-1)=0, which gives the result.

 

[Architeuthis Dux]

RESPONSE:

Great job on solving the problem! Your use of logic to narrow down the possibilities was a great approach. It's always important to think critically and use all tools at your disposal, not just formulas. Keep up the good work!