If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Algebra 1

### Course: Algebra 1>Unit 9

Lesson 3: Introduction to geometric sequences

# Intro to geometric sequences

Sal introduces geometric sequences and their main features, the initial term and the common ratio. Created by Sal Khan.

## Want to join the conversation?

• Can anyone explain to me why the geometric sequence follows the basic structure of an exponential function `[f(x) = a * r^x]` and why the arithmetic sequence follows the structure of a linear function `[f(x) = mx + b]`? I get that they are similar concepts but this intrigues me and I would like to understand the reasons behind this similarity. Thanks in advance. • The sequences are in fact linear and exponential functions, but with the domain restricted to positive integers.

The "standard form" of arithmetic sequences, "a(i) = a(1) + d(i-1)", as a function in the xy plane would be "f(x) = m(x-1) + b", where b = a(1) and m = d; which is a shift to the right by one unit of "f(x) = mx + b". This is because the common difference is not added to the first term in this sequence, while in "y = mx + b", m is added to f(x) at x = 1. When the sequence is given as "a(j) = a(1) + dj" (i.e. the common difference is added to the first term) it translates directly to "y = mx + b" with y = 0*x + b when x=0).
Similarly, the 1st term of a geometric sequence is in general independent of the common ratio. So the formula should be "a(i) = a(1)*r(i-1)" (shifted to the right), whereas as a function of real numbers an exponential is "y = (initial value)*r^x"

(Good question).
• Wouldn't the first table be incorrect ( in terms of the jump).
On the second jump, you would do 0.6 times 120 + 120 because 120 is the jump and, 06 times 120 is after the bounce? • Geometric sequences differ from arithmetic sequences. In geometric sequences, to get from one term to another, you multiply, not add. So if the first term is 120, and the "distance" (number to multiply other number by) is 0.6, the second term would be 72, the third would be 43.2, and so on.
• Sal, I'm more confused than when I started. This video is convoluted. • every sequence usually begins with "a1", why we can't begin with "a0" ? I studied the definition of the sequences as functions, that says a sequence is a function with the positive integer as your domain, and the real numbers as your counter domain. Why we can't use the 0 in this domain too? • You will find that sequences don't always start with a(1). Back here where math is simpler, we do often talk about a(1) as the first item.
It is really a semantics thing. If you mean that the a(1) is your first term, then you cannot have a zero term. Our terms are discrete.
However, you can define your first term as a(0) in the same way that in a computer array, the first element is the 0th item. It is usually easier for humans to keep track if the first item is called the n = 1 item.
When we use sequences to match or model actual occurrences, it can get pretty interesting. In the case that Sal is modeling, the first thing that happens is slightly different, so we call it "0"

Whichever way you start numbering, it is always important to check that your formula for the sequence actually ends up with the sequence that you want. If you start at a(1), you will usually need to have your exponent as the expression `n-1` to match the sequence that you are given.
On the other hand, if your sequence starts with a(0), you will often find your exponent needs to be `n` in order for your initial value to be correct for building the sequence you want.
• Why did we start at zero bounces instead of 1. In arithmetic we start at 1? • Yeah, I would say Sal overthought it a bit...

Obviously, there's only one jump and the first bounce has to occur after that jump.

– – –

So, after her initial (and only) jump Anne reaches a point where the bungee rope will cease extending and start retracting, which I would say counts as her 1st bounce.
At this point the bungee rope measures a length of 120 ft.

Then she is pulled upwards for a couple of seconds until gravity again gets the upper hand and pulls her down to her 2nd bounce, at which point the length of the rope is 60% of the length it measured at the previous bounce, i.e. 120 × 0.6 = 72 ft.

Then at her 3rd bounce, the length of the rope would be
120 × 0.6² = 43.2 ft.

Following the same logic, the length of the rope at Anne's 12th bounce would be 120 × 0.6¹¹ ≈ 0.435 ft.
• At sal multiplies 90 by -1/3. Why is it only mulitplication in geometric sequences but not divison? • when Sal introduced "n", why was it introduced? and does the letter stand for something? I got lost once that "n" was introduced. • n is the placement number or position of the term you want to find. The first term is always n=1, the second term is n=2, the third term is n=3 and so on. Therefore, this is not the value of the term itself but instead the place it has in the geometric sequence.

Saying "the nth term" means you can calculate the value in position n, allowing you to find any number in the sequence. You use n in the general formula of a geometric sequence and replace it with a number when you want to find the term in a certain position.

For example, if you have the general formula Un = 100 x (2)^n-1, you can use this to find any number in the sequence.

U1 is the first term and U1 = 100 x (2)^1-1 = 100 x (2)^0 = 100 x 1 = 100
U14 is the fourteenth term and U14 = 100 x (2)^14-1 = 100 x (2)^13 = 819200

Hope this helps!   