What's the common ratio for the sequence ?

The common ratio is the number multiplied to get the next term.

Solve this equation for r to find the common ratio. Multiply both sides by ½, the same as dividing by 2.

Now get r all by its lonesome.

List the first four terms and the 10th term of a geometric sequence with a first term of 3 and a common ratio of .

Our first term is 3, so a₁ = 3. Multiply the first term by the common ratio, , to get the second term.

Continue this process like a boss to find the third and fourth terms.

We need to find a rule for the sequence. Use the general form and plug in our first term and common ratio.

a_n = a₁(r)^{n – 1}

Substitute 10 for n in order to find the 10th term.

So our sequence looks a little somethin' somethin' like this:

How many terms exist in the sequence 1, 2, 4, … , 131,072?

Crank out the common ratio, first term, and last term of the sequence.

r = 2
a₁ = 1
a_n = 131,072

Use the information you've gathered and the general rule of a geometric sequence to create an equation with one variable, n. That's our total number of terms.

a_n = a₁(r)^{n – 1}

131,072 = 1(2)^{n – 1}

Solve the equation you've created for n.

131,072 = (2)^{n – 1}

Rewrite the equation as a logarithm with a base of 2 to solve for n using a calculator.

log₂(131,072) = n – 1
17 = n – 1
n = 18

Guess and check works as well if you're not into the whole logarithm scene.

2^{n – 1} = ?
2^{16 – 1} = 2¹⁵ = 32,768
2^{17 – 1} = 2¹⁶ = 65,536
2^{18 – 1} = 2¹⁷ = 131,072

n = 18

Either way, it turns out we've got 18 terms in the sequence.