Porisma

Worked solution

Relate three consecutive geometric terms

Suppose x,y,zx,y,z are three consecutive terms of a GP with common ratio rr. Write an equation relating x,y,zx,y,z.

CJC · Sequences and Series · Unnumbered GP mean question, page 16

Answer

y2=xzy^2=xz. If x,y,zx,y,z are all nonnegative, y=xzy=\sqrt{xz} is the geometric mean of xx and zz.

Compare your reasoning

Equate consecutive ratios

yx=zy=r⟹y2=xz\frac yx=\frac zy=r\quad\Longrightarrow\quad y^2=xz

GP terms in this lecture are nonzero, so the ratios are defined. The source uses the resulting identity to solve GP problems.

Keep the sign condition for the square root

y≥0⟹y=xzy\ge0\quad\Longrightarrow\quad y=\sqrt{xz}

The printed geometric-mean statement assumes all three terms are nonnegative. Without that assumption, y²=xz does not select the positive square root.

How did this question go?

Common mistakes

Do not discard a possible negative middle term when only y²=xz is known. Zero terms are excluded by the source GP definition.