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# Intro to geometric sequences (advanced)

Sal introduces geometric sequences and gives a few examples. Notation used in this video is relatively advanced. Created by Sal Khan.

## Want to join the conversation?

• How do you find the common ratio?
• You divide the 2nd term by the first term.
• When are geometric sequences used in real-life situations?
• After sequences you will learn about series, which are just the sums of sequences. We can use them to approximate complicated function values using simpler polynomials in numerical analysis, which is all about using calculus techniques with computers. Here is the section from Khan's integral calculus section that introduces the concepts.

Some of the algorithms in your calculator and computer might use these techniques to approximate complicated expressions involving cosx, sinx, e^x etc. So right there is a real life situation. They are also used a lot in electrical engineering.

Remember, there is no "learn this and then forget it" in math. Everything you learn is preparation for what is coming.

Here is a discussion related to your question:
• When Sal was defining the non-geometric sequence explicitly, he wrote,
a sub n with n=1 to infinity with a sub n = n!
I didn't really get what factorial is?
• A factorial is just a product of all integers(positive) less than or equal to 'n'
i.e.
n! = n x (n-1) x (n-2) x..........3x2x1
For example , lets say n=5, so n! i.e. 5! will be
5 x (5-1) x (5-2) x (5-3) x (5-4) = 5x4x3x2x1 = 120

if n=100 then 100! will be 100x99x....x3x2x1
• So, this tells you how to move forward, while using the sequence formula, but how do you go backwards? example: The 10th term in a geometric sequence is 0.78125, and the common ratio is -0.5. Find the first term in this geometric sequence.
• The answer Varun gives is slightly wrong, because of rounding, and also sign!
You divide 0.78125 by (-0.5^9) not a rounded version of that number - -0.5^9 is -0.001953125 but even if you entered all the digits, the dividing by the factored number will always be more accurate!

Such that 0.78125/(-0.5^9) = -400 = a

Therefore the first term is actually 400 if we apply the formula ar^n-1 => -400(-0.5^0) = 400
We can prove it by calculating the sequence up to the tenth term, but for briefness let's confirm that a(r^9) = 0.78125:

10th term = -400(-0.5^9) = 0.78125 exactly.

Whereas if a = 400.641 - then the 10th term would therefore be 400.641(-0.5^9) = -0.782501953125 - which is clearly NOT the correct result.

The correct answer for the first term in that geometric sequence is exactly 400.
• Is geometric sequence an exponential function?
• Yes, and the domain is restricted to positive integers.
• why is the name "geometric"??
• It is quite difficult to explain but here's the easiest explanation (well at least my attempt to do so):
Geometry (in arithmetic) means "the theory of numbers", since r varies according to n, and is therefore not a fixed number, we are here in a theory.
(1 vote)
• At shouldn't "a sub n" equal a*n!
• Yes, but a equals 1. "a sub n" is not "a."

I hope this clarifies the video!
(1 vote)
• I need a quick reminder: what is a recursive vs explicit? I can't seem to find the video that Sal explained this, and I seem to have neglected to write it down. Thanks!
• Recursive means defining the current term in terms of the previous term. E.g., a(n) = a(n-1)+2.

Explicit means defining the current term in terms of the current index value of the sequence. E.g. a(n) = 3*4^n.

(Those were two different examples; they aren't equivalent sequences)

So if i asked you what is the 145th element of a particular sequence, using the explicit definition will make things a LOT easier. If instead you used the recursive formulation, you'd have to know what the 144th element is, which would require you to know the 143rd element, and so on.....which would take all day.
• If each of the numbers which I'm gonna post below is divided by 2, does that mean that the sequence is geometric? Because I'm sure that's it's not arithmetic! Or maybe it's not one of those two. . . I can't tell:

1.6, 0.8, 0.4, 0.2. . .