![]() It's gonna get us right over there and then finally we take this last term and you multiply it by negative r, what do you get? You get, negative a times r to the n. Minus two times negative r is gonna give us this. Negative r is going to be, let me put subtraction signs, it's going to be negative a times r to the n minus one power. And we would keep going all the way to the term before this times negative r. And just to be clear what's going on, that's that term times negative r. Then, if I multiply ar times negative r that's going to be negative ar squared. Term, a times negative r, that's going to be negative ar. That's the equivalent of multiplying negative times the sum. I'm just gonna multiply everyone of these terms by negative r. So if you multiply this times negative r. So what is this going to be equal to? This is going to be equal to, well if you multiply a times negative r, we will get negative ar. Just add these two things and you'll see that itĬleans this thing up nicely. Actually, let's just multiply negative r. We're gonna take the r times that sum, r times the sum of the first nth terms. 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 Positive three over 256.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. Negatives together, so that's going to give Negative times itself four times, or I'm multiplying four It's going to be positive because I'm multiplying a Squared is four to the fourth so it's 16 times 16 is 256. let's see, four squared is 16, so four squared times four let's see, one to the one fourth is- oh, one to the fourth power is just one, and then four to the fourth power. It's gonna be a positive value so it's gonna be three times. ![]() Multiplying the negative an even number of times so It to an even power so it's going to give us a positive value since we're gonna be Times negative one fourth to the fourth power. ![]() Times negative one fourth to the five minus one power. I or a place with a five is going to be equal to three So given that, what is A sub five, the fifth term in the sequence? So pause the video and try to figure out what is A subscript five? Alright, well, we can Times negative one fourth to the I minus one power. Tell us that the Ith term is going to be equal to three Sequence A sub I is defined by the formula and so they
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