Intro to arithmetic progressions
- [Voiceover] All right, we're told that the arithmetic sequence a-sub-i is defined by the formula where the ith term in the sequence is going to be 4 plus 3 times i minus 1. What is a-sub-20? And so a-sub-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-sub-20, we just use this definition of the ith term. Everywhere we see an i, we would put a 20 in, so it's going to be 4 plus 3 times 20 minus 1, so once again, just to be clear, a-sub-20, where instead of a-sub-i, wherever we saw an i, we put a 20, and now we can just compute what this is going to be equal to. This is going to be equal to 4 plus 3 times 20 minus 1 is 19. 3 times 19, let's see, 3 times 19 is 57, right? It's 30 plus 27, yep. This is 57, and 4 plus 57 is equal to 61, so the 20th term in this arithmetic sequence is going to be 61. Let's do another one of these. And here, they've told us the arithmetic sequence a-sub-i is defined by the formula a-sub-1, they gave us the first term, and they say, every other term, so a-sub-i, they're defining it in terms of the previous terms, so a-sub-i is going to be a-sub-i minus 1 minus 2, so this is actually a recursive definition of our arithmetic sequence. Let's see what we can make of this, so a-sub-5, a-sub-5 is going to be equal to, we'll use this second line right here, a-sub-5 is going to be equal to a-sub-4 minus 2. Well, we don't know what a-sub-4 is just yet, so let's try to figure that out. So we could say that a-sub-4 is equal to, well if we use the second line again, it's going to be a-sub-3, minus 2. We still don't know what a-sub-3 is. I'll keep switching colors 'cause it looks nice. A-sub-3 is going to be equal to a-sub-3 minus 1, so a-sub-2 minus 2. We still don't know what a-sub-2 is, and so, we could write, a-sub-2 is equal to a-sub-2 minus 1, so that's a-sub-1 minus 2. Now, luckily we know what a-sub-1 is. A-sub-1 is negative 7. A-sub-1 is negative 7, so if this is negative 7, then a-sub-2 is negative 7 minus 2, which is equal to negative 9. Well that starts helping us out because if a-sub-2 is negative 9, if this is negative 9, then a-sub-3 is negative 9 minus 2, which is equal to negative 11. Well now that we know that a-sub-3 is negative 11, so this is negative 11, we could figure out a-sub-4 is negative 11 minus 2, which is equal to negative 13. And we're almost there. We know a-sub-4, the fourth term in this arithmetic sequence is negative 13, so, we can now, so if this is negative 13, a-sub-5 is going to be a-sub-4, which is negative 13 minus 2, which is equal to negative 15, so the fifth term in the sequence is negative 15. And we're all done.