To do it we're gonna thinkĪbout what r times the sum is. We want to come up withĪ nice clean formula for evaluating this and we're gonna use a little trick to do it. If we're on the nth term it's going to be ar to the n minus oneth power. So whatever term we're on the exponent is that term number minus one. Because notice, our first term is really ar to the zeroth power, second term is ar to the firsth power, third term is ar to the second power. We're gonna go all the way to the nth term and you might be tempted to say it's going toīe a times r to the nth power but we have to be careful here. It's going to be ar times r or ar squared. The second term times our common ratio again. Now, what's the third term going to be? Well, it's going to be S sub n is going to be equal to, you'll have your first term here, which is an a and then what's our second term going to be? This is a geometric series so it's going to be a times the common ratio. A formula for evaluatingĪ geometric series. Is using this information, coming up with a generalįormula for the sum of the first n terms. We're going to use a notation S sub n to denote the sum of first. We also know that it's aįinite geometric series. For example, we know that the first term of our geometric series is a. There are some things that we know about this geometric series. Let's say we are dealing with a geometric series. Try taking the sum of these series, and make a function for each of them, and then find a generic formula for all the diagonals if you're feeling brave!Ī tip i can give you, is to try to go from something you don't know to something you do know, the path between the two is "intuition".Īnd as a bonus, pascal's triangle has way more than just series, try exploring it and figuring out its properties, it's fascinating ! By doing so, you'll be building up your "intuition", I can guarantee it! if the greeks had known about it, they'd have built temples and revered it like a deity. I can't explain it properly but its super easy, so here how it goes : To make pascal's triangle you start with 1įor each consecutive row you add the number on the left and the right on the rows above to get your number, and a blank = 0. Here's a picture of pascal's triangle, and the "diagonals" are highlighted Practice helps build intuition, now for an endless amount of series to practice with I can only highly recommend pascal's triangle, and using its "diagonals" as series and trying to figure out the formula for each of them.
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