AP and GP

[Cookie98]

Hi,

I've been having a lot of problems with my math homework recently and have no idea how to solve it. Could anyone possibly help me?

1. Find such x and y for which the number x+y, x^2, y+2 are both in AP and in a GP.

2. The sum of three successive terms of a GP is 28. The numbers are first, second and fourth term of an AP. Fin the terms.

I will be really grateful if gives me at least one tip.

Thank you in advance.

 

[Deleted member 4993]

Hi,

I've been having a lot of problems with my math homework recently and have no idea how to solve it. Could anyone possibly help me?

1. Find such x and y for which the number x+y, x^2, y+2 are both in AP and in a GP.

2. The sum of three successive terms of a GP is 28. The numbers are first, second and fourth term of an AP. Fin the terms.

I will be really grateful if gives me at least one tip.

Thank you in advance.

What is the relationship between 3 consecutive terms in AP?

a_n+2 - a_n+1 = a_n+1 - a_n

similarly,

What is the relationship between 3 consecutive terms in GP?

a_n+2 / a_n+1 = a_n+1 / a_n

Now continue....

 

[Cookie98]

What is the relationship between 3 consecutive terms in AP?

a_n+2 - a_n+1 = a_n+1 - a_n

similarly,

What is the relationship between 3 consecutive terms in GP?

a_n+2 / a_n+1 = a_n+1 / a_n

Now continue....

I know that these are relationships, but when I try to solve the equation:

{y+2-x^2=x^2-(x+y)
{y+2/x^2=x^2/x+y

I'm just unable to solve it...

 

[stapel]

1. Find such x and y for which the number x+y, x^2, y+2 are both in AP and in a GP.

I know that these are relationships, but when I try to solve the equation:

{y+2-x^2=x^2-(x+y)
{y+2/x^2=x^2/x+y

I'm just unable to solve it...

You might be almost to the solution, so please reply showing what you've done so far.

2. The sum of three successive terms of a GP is 28. The numbers are first, second and fourth term of an AP. Fin the terms.

Apply the definitions for this exercise, too, letting "a" stand for the first term of each of the progressions.

. . .by definition of "geometric progression" with common ratio "r":

. . . . .a + ra + r²a = 28

. . .by definition of "arithmetic progression" with common difference "d":

. . . . .a, a + d = ra, a + 2d, a + 3d = r²a

From the AP, we can plug into the GP:

. . . . .a + a + d + a + 3d = 28

. . . . .3a + 4d = 28

We also know, again by definition, the following:

. . . . .(ra)/a = (r²a)/(ra)

Plugging in from the AP, we get:

. . . . .(a + d)/a = (a + 3d)/(a + d)

. . . . .a² + 2ad + d² = a² + 3ad

. . . . .d² = ad

What then must be the value of "d"?