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Hopefully, you have mastered the concepts in the previous lesson on Arithmetic Sequences before you proceed with this lesson. Definition: A geometric sequence is an ordered list of numbers separated by commas generated by multiplying the previous term by a constant number (r). "r" may be an integer or a fraction or decimal. When "r" is a positive integer, the numbers increase. When "r" is a positive fraction (0 < r <1), the numbers decrease. So...the terms in a Geometric Sequence may increase, sometimes called "growth", or may decrease, referred to as "decay" in the real world. When "r" is a negative value, the terms alternate between positive and negative. Since the terms are being multiplied the values tend to get very large (or small) quickly. The sequence {1,3,9,27,...} was mentioned in the previous lesson as NOT arithmetic. Actually, it is a sample of a geometric sequence. Note: each subsequent term is 3 times the previous term so "r" = 3. If "r" is not apparent it may be found by dividing two consecutive terms, in the right order. Other examples of Geometric Sequences are: {3,6,12,24,48,...} where "r" = 2 since 24/12 = 2.{10,20,30,40,...} is NOT a Geometric Sequence since there is not a constant "r" value. The formula for working with a Geometric Sequence differs from that used for an Arithmetic Sequence... The best way to explain these concepts is with a real world problem... Example: In a strep throat outbreak, the first 3 days of the infection period reported 5, 15, and 45 cases respectively. If the growth increases geometrically, how many people will be newly infected on the 7th day??? (New cases only, not total) Solution: We need to find all the values required by the formula above. r = 45/15 = 3... We are asked to find A7 so n - 1 = 6... Substituting into the formula we get a7 = 5 * 3 6 = 3,645 Note: Transmission of some communicable diseases follows a geometric model. These events are known as Epidemics. You might try this problem for yourself to see if you are following so far! The stock price of a modem manufacturer fell geometrically from $128 to $96 to $72 during the first 3 years that the stock traded. If this trend continues, what will each share of stock be worth in the fifth (5th) year??? A Geometric Series (denoted Sn) resembles a sequence except that the commas are replaced by "+". Geometric Series are really Geometric Sums. Note: Sums of terms in a sequence are considered to be "partial sums" since they are the sum of a specified number of terms. It is not possible to find a sum for an infinite number of terms. An exception would be when "r" is a fraction and the terms are decreasing instead of increasing. (Hence, the notion of "Limit".) There is a formula for finding partial sums for geometric sequences...
Consider {6,18,54,...}. Find S12, ie. find the sum of the first 12 terms. Substituting into the formula we get S12 = (6* (1 - 312)/(1-3) Better get out the calculator and be careful keying in the numbers. Expect quite a large answer since 312 is enormous. If everything was entered correctly, the answer should be 1,594,320.  Sigma Notation is a symbolic way to request a partial sum.Here is a sample problem and solution:  The problem is asking for the sum of the terms from #1 through #8 generated by the rule 4N. We must find all the values required for the execution of the formula. So... Now we have all the necessary values to substitute and... |