Geometric Sequences with a Surd Ratio

Finding the ratio

A geometric sequence multiplies by the same number each time, the common ratio. Divide any term by the one before to find it.

In $\sqrt{2}$√2, 2, $2\sqrt{2}$2√2, 4, … the ratio is $2÷\sqrt{2}=\sqrt{2}$2 ÷ √2 = √2, and $2\sqrt{2}÷2=\sqrt{2}$2√2 ÷ 2 = √2 confirms it. A surd ratio works exactly like a whole-number one.

Every second term

$\sqrt{2}\times \sqrt{2}=2$√2 × √2 = 2, so multiplying by $\sqrt{2}$√2 twice multiplies by 2. Every second term is twice the term two places before it: 4, then 8, then 16.

The terms in between carry the $\sqrt{2}$√2: $4\sqrt{2}$4√2, then $8\sqrt{2}$8√2.

The nth term

The nth term is $a\times r^{n-1}$a × rⁿ⁻¹. Here $a=\sqrt{2}$a = √2 and $r=\sqrt{2}$r = √2, so it is $\sqrt{2}\times {\left(\sqrt{2}\right)}^{n-1}={\left(\sqrt{2}\right)}^n$√2 × (√2)ⁿ⁻¹ = (√2)ⁿ.

An even power of $\sqrt{2}$√2 is a power of 2: ${\left(\sqrt{2}\right)}^{10}={\left({\left(\sqrt{2}\right)}^2\right)}^5=2^5=32$(√2)¹⁰ = ((√2)²)⁵ = 2⁵ = 32. An odd power leaves one $\sqrt{2}$√2 over: ${\left(\sqrt{2}\right)}^7=2^3\times \sqrt{2}=8\sqrt{2}$(√2)⁷ = 2³ × √2 = 8√2.

A missing term

If 1, x and 3 are consecutive terms, the ratio from 1 to x equals the ratio from x to 3: $x/1=3/x$x / 1 = 3 / x. Multiplying both sides by x gives $x^2=3$x² = 3.

$x=\sqrt{3}$x = √3 gives 1, $\sqrt{3}$√3, 3, multiplying by $\sqrt{3}$√3 each time. $x=-\sqrt{3}$x = −√3 works too: 1, $-\sqrt{3}$−√3, 3 multiplies by $-\sqrt{3}$−√3. A question says x > 0 when it wants only the positive one.

Ratios less than 1

The sequence 3, $\sqrt{3}$√3, 1, … divides by $\sqrt{3}$√3 each time, so its ratio is $\frac{1}{\sqrt{3}}$1/√3. Rationalized, that is $\frac{\sqrt{3}}{3}$√3/3: multiply top and bottom by $\sqrt{3}$√3.

The next term is $1\times \frac{\sqrt{3}}{3}=\frac{\sqrt{3}}{3}$1 × √3/3 = √3/3.

Adding the terms

Collect the whole numbers and the surds separately. $\sqrt{2}+2+2\sqrt{2}+4=\left(2+4\right)+\left(\sqrt{2}+2\sqrt{2}\right)=6+3\sqrt{2}$√2 + 2 + 2√2 + 4 = (2 + 4) + (√2 + 2√2) = 6 + 3√2.

The usual slips

Subtracting to find the ratio. $2-\sqrt{2}$2 − √2 is the gap between two terms; a geometric sequence divides.

Counting each $\sqrt{2}$√2 as a 2. ${\left(\sqrt{2}\right)}^{10}$(√2)¹⁰ has ten factors of $\sqrt{2}$√2, which is only five factors of 2: 32, not $2^{10}=1024$2¹⁰ = 1024.

Squaring instead of rooting. $x^2=3$x² = 3 means $x=\sqrt{3}$x = √3, not 9.