![]() Then, because | r | < 1, we can find the sum of the infinite series.Ģ0 A financial example You want to save money by setting aside a 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on. (Why?)įind the sum of the alternating geometric series It may help to calculate the first couple of terms to verify the first term and r. If r < 0, we have something called an alternating infinite series. This is called an infinite series, and we can find the sum only in this particular case.Īn infinite series may be noted using summation notation. If | r | < 1, then we can determine the sum of the entire geometric series. Here’s the bottom line: Check it with our sequence: = 6560 How long did that take? Want a shortcut? Not surprisingly, there are formulas. When we did partial sums of arithmetic sequences, those were also series.)ġ5 Partial sums Add the first 8 terms of our sequence (And here’s a free vocabulary word: when we add the terms of a sequence, we call it a series. r = -2 a1 = 5/4 Once we have r and a1, we can find the equation. 5 = a1 (-2)2 5 = a1 (4) a1 = 5/4ġ3 Use it again If the third term of a geometric sequence is 5 and the sixth term is -40, find the eighth term. 40 = a1(r)5 5 = a1(r )2 -8 = r3 and r = -2ġ2 Use it again If the third term of a geometric sequence is 5 and the sixth term is -40, find the eighth term. Once we have the equations, we can find r. Now what? -40 = a1(r) = a1(r)5 5 = a1(r) = a1(r)2ġ1 Use it again If the third term of a geometric sequence is 5 and the sixth term is -40, find the eighth term. Like we did with arithmetic sequences, we start by writing the equations. Like with arithmetic sequences, we need the first term and the change between terms. Explicit: an = a1rn-1 (In other words, find the nth term by multiplying a1 by r and do that (n-1) times.)ħ Geometric sequences Recursive: a1 an = an-1r so, r = an/an-1 Explicit:Īn = a1rn-1 How does these compare to the formulas for an arithmetic sequence?Ĩ Use it Find the 10th term of our POD sequence an = 2(3)n-1ĩ Use it Find the 10th term of our POD sequence an = a1(3)n-1ġ0 Use it again If the third term of a geometric sequence is 5 and the sixth term is -40, find the eighth term. Recursive: a1 an = an-1r so, r = an/an-1 What would the explicit formula be?Ħ Geometric sequences If the pattern between terms in a sequence is a common ratio, then it is a geometric sequence. ![]() ![]() Each term is 3 times the previous term.ĥ Geometric sequences If the pattern between terms in a sequence is a common ratio, then it is a geometric sequence. 3 POD preview Give the first 5 terms of the sequence forĪn = a1(3)n if a1 =2 Is this formula recursive or explicit? What is the pattern in this sequence? How do we know?Ĥ POD preview Give the first 5 terms of the sequence forĪn = a1(3)n if a1 =2 2, 6, 18, 54, 162 This is an explicit formula. ![]()
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