Question

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?

Answer 1

Answer:

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

Step-by-step explanation:

A number when added to $x,y,z$ that yields consecutive terms of a geometric sequence is an unknown number $t\in \mathbb{Z}$

Given

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

We know

$\alpha _1 = 1+t$

$\alpha _2 = 7+t$

$\alpha _3 = 15+t$

Recall that a geometric sequence is in the form

$\boxed{a_n = a_1 \cdot r^{n-1}}$

Therefore, once  $\alpha_1, \alpha_2, \alpha_1$ are consecutive terms,

$15+t = (1+t) r^{3-1} \implies 15+t = (1+t) r^2$

To find the ratio, for

$\dots a_{k-1}, a_k, a_{k+1} \dots$

we know

$\dfrac{a_k}{a_{k-1}} =\dfrac{a_k}{a_{k-1}} =r$

Therefore,

$\dfrac{(7+t)}{(1+t)} =\dfrac{(15+t)}{(7+t)} \implies (7+t)^2 = (15+t)(1+t)$

$\implies 49+14t+t^2=15+16t+t^2 \implies -2t=-34 \implies t=17$

The ratio is therefore

$r=\dfrac{4}{3}$

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