In this lesson, our instructor Vincent Selhorst-Jones teaches geometric sequences and series. He discusses geometric sequences, geometric series, form for the nth term, formula for the geometric series, and the infinite geometric series.
A sequence is geometric if every term in the sequence can be given by multiplying the previous term by some constant number r:
an = r·an−1.
We call r the common ratio (since r = [(an)/(an−1)]). Every "step" in the sequence multiplies by the same number. The number can be anything, so long as it is always the same for each step.
The formula for the nth term (general term) of a geometric sequence is
an = rn−1 ·a1.
To find the formula for the general term of a geometric sequence, we only need to figure out its first term (a1) and the common ratio (r).
We can use the following formula to calculate the value of a geometric series. Given any geometric sequence a1, a2, a3, …, the sum of the first n terms (the nth partial sum) is
Sn = a1 ·
We can find the partial sum (Sn) by only knowing the first term (a1), the common ratio (r), and how many terms are being added together (n). [Caution: Be careful to pay attention to how many terms there are in the series.
It can be easy to get the value of n confused and accidentally think it is 1 higher or 1 lower than it really is.]
Unlike an arithmetic series, we can also consider an infinite series when working with a geometric sequence. If |r| < 1, then
S∞ = a1 ·
Geometric Sequences & Series
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.