The first three terms of a geometric sequence of positive numbers are x, y, and z, in that… — ACT

The first three terms of a geometric sequence of positive numbers are $x$ , $y$ , and $z$ , in that order. An arithmetic sequence has first three terms $x$ , $y$ , and $z - 4$ , in that order. If $x = 4$ , what is the value of $y$ ?

A
2
B
4
C
6
8Answer
E
16

Answer

The correct answer is 8. By defining the geometric sequence terms as

4, y, z

, we have the property

y^2 = 4z

. For the arithmetic sequence

4, y, z-4

, the common difference property gives

y - 4 = z - 4 - y

, which simplifies to

2y = z

. Substituting

z = 2y

into the first equation yields

y^2 = 8y

. Since all terms must be positive, dividing by

y

gives

y = 8

Step-by-Step Solution

Set up the equation for the geometric sequence.

y^2 = 4z

Since

4, y, z

is a geometric sequence, the ratio of consecutive terms must be equal:

\frac{y}{4} = \frac{z}{y}

, which simplifies to

y^2 = 4z

Set up the equation for the arithmetic sequence.

2y = z

Since

4, y, z - 4

is an arithmetic sequence, the difference between consecutive terms must be equal:

y - 4 = (z - 4) - y

. Simplifying this gives

2y = z

Substitute the arithmetic equation into the geometric equation.

y^2 = 8y

Substituting

z = 2y

into

y^2 = 4z

yields

y^2 = 4(2y) = 8y

Solve for the variable

y

y = 8

Rearranging the equation gives

y^2 - 8y = 0

, which factors as

y(y - 8) = 0

. Since the sequence consists of positive numbers,

y

must be positive, so

y = 8

Key Concept

Relating arithmetic and geometric sequence properties to solve a system of non-linear equations

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