yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Using arithmetic sequences formulas | Mathematics I | High School Math | Khan Academy


2m read
·Nov 11, 2024

All right, we're told that the arithmetic sequence ( a_i ) is defined by the formula where the ( i )-th term in the sequence is going to be ( 4 + 3 \cdot (i - 1) ). What is ( a_{20} )?

So, ( a_{20} ) is the 20th term in the sequence, and I encourage you to pause the video and figure out what is the 20th term. Well, we can just think about it like this: ( a_{20} ), we just use this definition of the ( i )-th term. Everywhere we see an ( i ), we would put a 20 in.

So, it's going to be ( 4 + 3 \cdot 20 - 1 ). So once again, just to be clear, ( a_{20} ), where instead of ( a_i ), wherever we saw an ( i ), we put a 20. Now we can just compute what this is going to be equal to.

This is going to be equal to ( 4 + 3 \cdot 20 - 1 ).

Let's see, ( 3 \cdot 20 ) is 60. So, this is ( 4 + 60 - 1 ), which equals ( 4 + 60 - 1 = 63 ). Thus, the 20th term in this arithmetic sequence is going to be 63.

Let's do another one of these. Here, they've told us the arithmetic sequence ( a_i ) is defined by the formula ( a_1 ). They give us the first term and say every other term, so ( a_i ), they're defining it in terms of the previous term.

So, ( a_i ) is going to be ( a_{i - 1} - 2 ). This is actually a recursive definition of our arithmetic sequence. Let's see what we can make of this.

So, ( a_5 ) is going to be equal to... we'll use this second line right here. ( a_5 ) is going to be equal to ( a_4 - 2 ). Well, we don't know what ( a_4 ) is just yet, so let's try to figure that out.

So, we could say that ( a_4 ) is equal to... well, if we use the second line again, it's going to be ( a_{3} - 2 ). We still don't know what ( a_{3} ) is. I'll keep switching colors because it looks nice.

( a_3 ) is going to be equal to ( a_{2} - 2 ). We still don't know what ( a_{2} ) is. So we could write ( a_2 = a_{1} - 2 ). Now, luckily, we know what ( a_1 ) is. ( a_1 ) is -7.

So if ( a_1 ) is -7, then ( a_2 = -7 - 2 ), which is equal to -9. Well, that starts helping us out because if ( a_2 ) is -9, then ( a_3 = -9 - 2 ), which is equal to -11.

Well, now that we know that ( a_3 = -11 ), we can figure out ( a_4 = -11 - 2 ), which is equal to -13.

And we're almost there! We know ( a_4 ). The fourth term in this arithmetic sequence is -13, so we can now... if this is -13, ( a_5 ) is going to be ( a_4 ), which is -13 - 2, which is equal to -15.

So the fifth term in the sequence is -15, and we're all done.

More Articles

View All
how I got rid of my ACNE after 8 years - ONLY thing worked
If drinking more water, exercising more, sleeping more, reducing stress, or generally the tips that people give you on YouTube didn’t work for you then this video is for you. Hi, guys, it’s me Ruri. Today, I’m back with another very requested video. Just …
Balaji Srinivasan at Startup School 2013
I can talk about white combinator. I guess you guys all know about that. Uh, let me introduce myself briefly while, uh, things are loading here. So, uh, my name is Bology S. Boson. Um, there’s actually 12 people with my same first and last name in the Bay…
15 Signs You Get Played By Others
Do you feel like you’re always the third wheel? The one who is easily taken advantage of, or the one whose opinions don’t matter? Well, in this video, we’ll explore why you always get played and what you can do to change that. From lacking presence and fa…
Diode graphical solution
Now I want to use a diode in a circuit and we’ll see how we, uh, solve circuits that include these nonlinear diodes in them. So I have a circuit here with a battery and a resistor and a diode here, and it’s going to be a special kind; it’s going to be an …
The aggregate production function and growth | APⓇ Macroeconomics | Khan Academy
In a previous video, we have introduced the idea of an aggregate production function, which is a fancy way for a mathematical model that an economist might use to tie the factors of production in an economy to the actual aggregate output of an economy. Th…
Earthships: A House Made From Beer Cans Sparks a Movement | Short Film Showcase
People look at this and call it a Mad Max compound. What the heck? These people live like this and a bunch of dirty hippies that don’t know how to clean up the land. Wow, it’s weird. They call it trashy. The world is not going to build weird houses, but y…