Geometric and arithmetic series

[Government$]

Homework Statement

If a,b,c, are at the same time fifth, seventh and thirty seventh member of arithmetic and geometric progression then $a^{b-c}b^{c-a}c^{a-b}$ is:

The Attempt at a Solution

I tried solving system of equations but i have four unknown. I was able to reduce it to on unknown.

12r^32 - 32r^12 + 20=0 where r is common ration in geometric series. I have no idea how to solve this.

Maybe trying to solve the system isn't a way to go?

 

[Government$]

Wel its evident that two solutions are 1 and - 1 but what kind of geometric progression is with r=1 or r=-1?

 

[Yukoel]

Homework Statement

If a,b,c, are at the same time fifth, seventh and thirty seventh member of arithmetic and geometric progression then $a^{b-c}b^{c-a}c^{a-b}$ is:

The Attempt at a Solution

I tried solving system of equations but i have four unknown. I was able to reduce it to on unknown.

12r^32 - 32r^12 + 20=0 where r is common ration in geometric series. I have no idea how to solve this.

Maybe trying to solve the system isn't a way to go?

Hello Government$
Do you mean that a,b,and c are parts of an arithemtico-geometric sequence(As in saying that they can be represented as the product of corresponding terms of an arithmetic and geometric series) or implying that there exist separate (not to be sure) arithmetic and geometric progressions satisfying the condition?
Regards
Yukoel

 

[Government$]

As i have understood it there exist separate arithmetic and separate geometric progression. This is a first time i hear of arithemtico-geometric series.

 

[Yukoel]

As i have understood it there exist separate arithmetic and separate geometric progression. This is a first time i hear of arithemtico-geometric series.

Hello,
Thanks for clarifying this. Well the way I can think of it is doesn't utilize finding the common difference and /or common ratio .Try writing them separately as nth(n=5,7,and 37 as given) terms of the Geometric and arithmetic sequence (Don't be disheartened by the number of unknowns :) ).Now look at the expression .In order to simplify it you might want to multiply the bases easily, by which sequence would you represent it(I mean a,b and c)? If you have had multiplied you might need to easily add the exponents. Which sequence's use makes it easier?
Regards
Yukoel

 

[Curious3141]

Homework Statement

If a,b,c, are at the same time fifth, seventh and thirty seventh member of arithmetic and geometric progression then $a^{b-c}b^{c-a}c^{a-b}$ is:

The Attempt at a Solution

I tried solving system of equations but i have four unknown. I was able to reduce it to on unknown.

12r^32 - 32r^12 + 20=0 where r is common ration in geometric series. I have no idea how to solve this.

Maybe trying to solve the system isn't a way to go?

This is a simple problem. You're told that a,b,c are particular terms of an arithmetic progression (A.P.) and a geometric progression (G.P.). So just use symbols to represent the first term and common difference of that A.P. and the first term and common ratio of the G.P. and express a,b,c both ways.

You're asked to evaluate an expression that's the product of powers of a, b and c. For the bases (e.g. a or b), use the G.P. representation. For the exponents (e.g. b-c), use the A.P. representation. Do the algebra using the laws of exponents and you'll be pleasantly surprised at what cancels out.