Our formula for the sum or for the sum of a finite geometric series. It's going to be a times one minus r to the n over one minus r. Of a drum roll here, S sub n is going to be equal to this divided by one minus r. Sum of our first n terms, we deserve a little bit It's going to be a times one minus r to the n. You factor that out, it's gonna be times one minus r is going to be equal to and on the right hand side we can actually factor out an a. Let's see, we can factor out an S sub n on the left hand side.
Now we can just solve for S sub n and we have our formula, It's going to be a minus a times r to the nth power.
All we're gonna have left with is negative ar to the n. The a sits there but everything else, except for this last thing, is going to cancel out. On the left hand side we get, S sub n minus r times S sub n and on the right hand side we have something very cool happening. Now what's interesting here is we can add up the left side and we canĪdd up the right hand sides. You multiply this times the negative, you get the negative a and then r to the n minus one times r, or times r to the first, well this is going to be r to the n. 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
ALGEBRA 2 GEOMETRIC SEQUENCES FORMULA SERIES
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 \) so there is no common ratio.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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