Find a value for x such that 3+x, 4+x, and 5+x are consecutive terms in a geometric sequence.?

Question

Find a value for x such that 3+x, 4+x, and 5+x are consecutive terms in a geometric sequence.
I did (5+x)/(4+x)=(4+x)/(3+x) to get x, but the x's all cancel out.
Let me know what I'm doing wrong?

Answer 1

(5+x)/(4+x)=(4+x)/(3+x) is correct.

Do cross multiplication:
15+8x+x^2 = 16+8x+x^2
which leads to 15 = 16

Therefore, there is no solution.

Answer 2

A geometric sequence is a sequence that has a common ratio (r). For this to be a geometric sequence (t2)/(t1) = r and (t3)/(t2) = r. Therefore, (t2)/(t1) = (t3)/(t2). We can plug the terms into this equation now to solve for x (t1 = 3+x; t2 = 4+x; t3 = 5+x).

(4+x)/(3+x) = (5+x)/(4+x) "cross multiply"
(4+x)(4+x) = (5+x)(3+x) "expand and simplify"
16 +8x +x^2 = 15 + 8x + x^2 "group like terms"
16 = 15 therefore no solution

This is an arithmetic sequence since these consecutive integers have a common difference of one and no common ratio.

Answer 3

No matter what value you set x equal to, x+3, x+4, and x+5 will be consecutive integers. The common difference is the constant 1, and therfore the numbers are part of an arithmetic sequence. The reason your x's cancel out is that the problem has no solution.

Answer 4

You didn't do anything wrong. The x's cancel leaving 16 = 15. That means there is no solution. So the answer is that no such geometric sequence exists.

Answer 5

using

an = a1 * r^(n - 1)

a2 = (3 + x) * r^(2 - 1)
4 + x = (3 + x)r
r = (4 + x)/(3 + x)

a3 = (3 + x) * r^(3 - 1)
5 + x = (3 + x) * r^2
(5 + x)/(3 + x) = r^2
r = sqrt((5 + x)/(3 + x))
r = (sqrt((5 + x)(3 + x)))/(3 + x)

(4 + x)/(3 + x) = sqrt((5 + x)(3 + x))/(3 + x)

4 + x = sqrt((5 + x)(3 + x))

(4 + x)^2 = (5 + x)(3 + x)

(4 + x)(4 + x) = 15 + 5x + 3x + x^2
16 + 4x + 4x + x^2 = 15 + 8x + x^2
16 + 8x + x^2 = 15 + 8x + x^2
16 = 15

Something isn't working out right.

Answer 6

x-3 ,. x+ 3 ,., 3x- 3 are in G.P. then ( x+3)^3 = (x-3) ( 3x- 3) [ a,b,,c are G.P. then b^2 = ac ] x^2 + 6x + 9 = 3x^2 - 3x -9x + 9 or2x^2 =18x or x= 0 & x=9 so G.P.is -3 . 3 ,-3---------- and G.P. is --------- 6, 12, 24------------- answer [ a= 6 & r =2 ]