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For a geometric sequence, , for every positive integer k.

In the previous section, we learned about arithmetic sequences. Arithmetic sequences have a common difference. Example: 3, 8, 13, 18, 23, 28, . . . (Arithmetic Sequence).

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For a geometric sequence, , for every positive integer k.

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  1. In the previous section, we learned about arithmetic sequences. Arithmetic sequences have a common difference. Example: 3, 8, 13, 18, 23, 28, . . . (Arithmetic Sequence) Another type of sequence is a geometric sequence. The ratio of a term in a geometric sequence to its preceding term is always the same number. For example; 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, . . . For a geometric sequence, , for every positive integer k. General Term of an Arithmetic Sequence For a geometric sequence, the general term is: an = a1 r(n-1) , where a1 is the first term and r is the common ratio. A geometric sequence is a sequence in which the ratio between a term and its preceding term is a constant. The number, r, is called the common ratio for the geometric sequence. For the geometric sequence above, the common ratio is 2.

  2. The common ratio is, Note that we could have used the ratio between any two consecutive terms, i.e., The general term for that in a geometric sequence, an = a1 r(n-1) So for the geometric sequence above, an = (8) 4(n-1) Simplifying this expression even more: Your Turn Problem #1 Find the general term for the geometric sequence: Example 1. Find the general term for the geometric sequence: 8, 32, 128, 512, 2048, . . .

  3. The common ratio, Now we will use the general term formula of a geometric sequence, Your Turn Problem #2 Find the general term for the geometric sequence: 0.2, 0.04, 0.008, 0.0016,…

  4. For a geometric sequence, the formula for the general term can be used to find a desired term. Example 3. Find the 9th term for the geometric sequence, -4, -24, -144, -864, . . . Your Turn Problem #3 Find the 12th term for the geometric sequence: 1.3, 3.9, 11.7, 35.1, . . . For this sequence, the common ratio, r = (-24) ¸ (-4) = 6 So, an = (-4) 6(n–1) Since we want the 9th term, let n = 9. Thus, a9 = (-4) 6(9–1) = (-4) 68 = (-4) (1,679,616) = -6,718,464 Answer: The 9th term, a9 = -6,718,464 Answer: The 12th term, a12 = 230,291.1

  5. Adding the Terms of a Geometric Sequence To find the sum of an geometric sequence, we need to know the first term, a1, the common ratio, r, and the number of terms, n. Sum formula for the first n terms of a geometric sequence. or Note: There are many forms of the formula for the sum of a geometric sequence. The two forms above are commonly used. Using the first form of the finite sum formula with a1 = 6, n = 8, and r = 10, we get Answer: The sum of the first 8 terms is 66,666,666. Your Turn Problem #4 Find the sum of the first 12 terms of the geometric sequence, 3, -12, 48, -192, . . . Example 4. Find the sum of the first 8 terms of the geometric sequence, 6, 60, 600, 6000, . . . For this sum, n = 8 and r = 60 ¸ 6 = 10 Answer: The sum of the first 12 terms is S12 = -10,066,329

  6. The finite sum formula can be used with any geometric sequence even if the common ratio, r, is a fraction or a negative number. Example 5.Find the sum of the first 9 terms of the geometric sequence, 875, -175, 35, 7, . . . For this sequence, r = -175 ¸ 875 = -1/5 For this sum, n = 9 Using the finite sum formula with a1 = 875, n = 9, and r = -1/5, we get Answer: The sum of the first 9 terms, S9 = 729 522/3125 or 729.16704 Your Turn Problem #5 Find the sum of the first 8 terms of the geometric sequence, 144, 96, 64, 42 2/3, . . .

  7. For a geometric sequence, if the number of terms of to be added is unknown, we need to either count the number of terms or use the nth-term formula to find n. Example 6.Find 3 + 33 + 363 + 3,993 + . . . + 643,076,643 Now use the sum formula: (Solve by matching bases or using logs.) Answer: 707,384,307 Your Turn Problem #6 Find (-7) + (-14) + (-28) + (-56) + . . . + (-917,504) For this sum, we must first determine if this is a geometric sequence. To do this, determine if there is a common ratio between the terms. For this sequence, a2 ¸ a1 = 11, a3 ¸ a2 = 11, a4 ¸ a3 = 11. So, there is a common ratio, r = 11, which means that we are, in fact, working with a geometric sequence. To find the number of terms, n, use the nth term formula. Then use the sum formula with n, a1 = 3, and r = 11, Answer: r = 2 and n = 18. The sum is -1,835,001.

  8. Geometric Series Summation notation can be used to indicate the sum of a geometric sequence. When asked to evaluate a sum, write out at least the first three terms. Good idea but not necessary to calculate the last term, n. Example 7.Find Answer: 2,441,405 Your Turn Problem #7 We can calculate this sum using the finite sum formula with a1 = 5, r = 5, and n = 9. Recall n= 9 - 1+1 = 9. (upper limit – lower limit +1) Answer: a1 = -6, n = 12, and r = -6. The sum is 1,865,813,430.

  9. Example 8.Find Using the finite sum formula, with a1 = 27, r = 1/3, and n = 11 (13-3+1), we get: Answer: Your Turn Problem #8 Answer: The sum is –531,684.

  10. Infinite Geometric Series The sum of an infinite geometric sequence is give by the formula: Example 9.Find Your Turn Problem #9 Since |r|<1, we can use the infinite sum formula with a1 = ¾ and r = ¾ .

  11. Writing Repeating Decimals in Fractional Form A repeating decimal number can be written as the sum of terms from a geometric sequence. The common ratio, r, is 10-R, where R equals the number of digits that are repeated. Since 0<| r|<1, the infinite sum formula can be used to write the repeating decimal in fractional form. Note that the terms 0.63, 0.0063, 0.000063, . . . form a geometric sequence with a common ratio of r = 10 -2 = 1/100 = 0.01 (two digits, 6 and 3, are repeated so R=2). Your Turn Problem #10 Using the infinite sum formula with a1 = 0.63 and r = 0.01 , we get

  12. If part of a repeating decimal number does not repeat, then treat that non-repeating part as a terminating decimal when converting to fraction form. Note that the terms 0.057, 0.00057, 0.0000057, . . . form a geometric sequence with a common ratio of r = 10 -2 = 1/100 = 0.01 (two digits, 5 and 7, are repeated so R=2). Your Turn Problem #11 Using the infinite sum formula with a1 = 0.057 and r = 0.01 , we get The End 4-3-2007

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