If x = 1, y = 7, and z = 15, determine a number that when added to x, y, and z yields consecutive terms of a geometric sequence. What are the first three terms in the
geometric sequence?

Given:

x = 1, y = 7,z = 15x=1,y=7,z=15

A number is added to x, y, and z yields consecutive terms of a geometric sequence.

To find:

The number which is added to x, y, and z, then find the first three terms in the geometric sequence.

Solution:

Let the unknown number be k.

After adding k to x, y, and z, we get

x = 1+k, y = 7+k,z = 15+kx=1+k,y=7+k,z=15+k

These are the consecutive terms of a geometric sequence. So,

(7+k)^2=(1+k)(15+k)(7+k)

2

=(1+k)(15+k)

49+14k+k^2=15+k+15k+k^249+14k+k

2

=15+k+15k+k

2

49+14k=15+16k49+14k=15+16k

Isolate the variable terms.

49-15=16k-14k49−15=16k−14k

34=2k34=2k

\dfrac{34}{2}=k

2

34

=k

17=k17=k

The unknown number is 17.

Now,

x=1+kx=1+k

x=1+17x=1+17

x=18x=18

Similarly,

y=7+ky=7+k

y=7+17y=7+17

y=24y=24

And

z=15+kz=15+k

z=15+17z=15+17

z=32z=32

Therefore, the three terms in the geometric sequence are 18, 24, 32.