 Page Two Here is another infinite geometric sequence: 27, -9, 3, -1, ⅓, -1/9, 1/27, -1/81, 1/243 ... t1 = 27 r = -⅓ Let's start adding terms: S1 = 27 S2 = 27 + -9 = 18 S3 = 18 + 3 = 21 S4 = 21 + -1 = 20 S5 = 20 + ⅓ = 20.333 here we'll start using decimals rounded to 3 places S6 = 20.333 + -1/9 = 20.333 - .111 = 20.222 S7 = 20.222 + 1/27 = 20.222 + 0.037 = 20.259 S8 = 20.259 + -1/81 = 20.259 - 0.012 = 20.247 S9 = 20.247 + 1/243 = 20.247 + 0.004 = 20.251 We're still a long way from an infinite number of terms. However, it sort of looks like the sum is tending towards 20.25 The sum is getting closer and closer to 20.25, which it will reach if there are an infinite number of terms We can say this using limits: limn→∞Sn = 20.25 In other words: S∞ = 20.25 The reason that we were able to find a sum of an infinite number of terms is because the value of r was a fraction whose size was less than 1. Finding the sum of an infinite sequence the way we've been doing it involves adding more and more terms in the sequence until we can guess at what the sum seems to be approaching. There is a quicker way; there is a formula that will let us fing the sum of an infinite geometric sequence, as long as -1 < r < 1 |