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

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.