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Partial Sums for Geometric Series

Explore geometric series concepts including partial sums, sequences, and formulas through examples and applications. Learn how to solve problems involving geometric sequences and expand your understanding of related mathematical concepts.

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Partial Sums for Geometric Series

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  1. Partial Sums for Geometric Series Section 9.3

  2. If a pair of calculators can be linked and a file transferred from one calculator to the other in 20 s, how long will it be before everyone in a lecture hall of 250 students has the file? During the first time period, the file is transferred to one calculator; during the second time period, to two calculators; during the third time period, to four more calculators; and so on. The number of students who have the file doubles every 20 s. • To solve this problem, you must determine the minimum value of n for which Sn exceeds 250. The solution uses a partial sum of a geometric series. It requires the sum of a finite number of terms of the geometric sequence 1, 2, 4, 8, . . . .

  3. Example A • Consider the sequence 2, 6, 18, 54, . . . . • Find u15. • Graph the partial sums S1 through S15, and find the partial sum S15. The sequence is geometric with u1=2 and r=3. a. A recursive formula for the sequence is u1=2 and un =3un-1 where n ≥ 2. The sequence can also be defined explicitly as un=2(3)n-1. Substituting 15 for n into either equation gives u15= 9,565,938. b. Use your calculator to graph the partial sums. Use a table or trace the graph to find that S15 is 14,348,906.

  4. Geometric Series Formula • A ball is dropped from 180 cm above the floor. With each bounce, it rebounds to 65% of its previous height.

  5. Step 1: Use your calculator to find the first ten heights, with the initial drop height being the first term of the sequence. Then find the first ten partial sums of this sequence of heights: S1, S2, S3, . . . , S10. • Step 2: Create a scatter plot of points n, Sn and find a translated exponential equation to fit the data. • The equation will be in the form Sn = L - a • bn. (Hint: L is the long-run value of the partial sums.) • Step 3: Rewrite your equation from Step 2 in terms of n, a or u1, and r. Use algebraic techniques to write your explicit formula as a single rational expression.

  6. Step 4: Here is a way to arrive at an equivalent formula. Complete the derivation by filling in the blanks. • Step 5: Use your formula from Step 4 to find S10 for the bouncing ball. Then use it to find S10 for the geometric sequence 2, 6, 18, 54, . . . .

  7. Example B • Find S10 for the series 16, 24, 36, . . . . The first term, u1, is 16. The common ratio, r, is 1.5. The number of terms, n, is 10. Use the formula you developed in the investigation to calculate S10.

  8. Example C • Each day, the imaginary caterpillarsaurus eats 25% more leaves than it did the day before. • If a 30-day-old caterpillarsaurus has eaten 151,677 leaves in its brief lifetime, how many will it eat the next day?

  9. To solve this problem, you must find u31. The information in the problem tells you that r is • (1+ 0.25), or 1.25, and when n equals 30, Sn equals 151,677. • Substitute these values into the formula for Sn and solve for the unknown value, u1.

  10. Now you can write an explicit formula for the terms of the geometric sequence, un = 47(1.25)n-1. • Substitute 31 for n to find that on the 31st day, the caterpillarsaurus will eat 37,966 leaves.

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