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Chapter 11 Sec 3

Chapter 11 Sec 3. Geometric Sequences. Geometric Sequence. A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a constant r called the common ratio. Example 1.

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Chapter 11 Sec 3

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  1. Chapter 11 Sec 3 Geometric Sequences

  2. Geometric Sequence • A geometric sequenceis a sequence in which each term after the first is found by multiplying the previous term by a constant r called the common ratio.

  3. Example 1 Find the eighth term of a geometric sequence for which a1 = – 3 and r = – 2. an = a1 · r n – 1 a8 = (–3) · (–2)8 – 1 a8 = (–3) · (–128) a8 = 384

  4. Example 2 Write an equation for the n term of a geometric sequence 3, 12, 48, 192… an = a1 · rn – 1 an = (3) · (4) n – 1 So the equations is an = 3(4) n – 1

  5. Example 3 Find the tenth term of a geometric sequence for which a4 = 108 and r = 3. an = a1 · r n – 1 a4 = a1 · (3)4 – 1 108 = a1 · (3)3 108 = 27a1 4 = a1 an = a1 · r n – 1 a10 = 4· (3)10 – 1 a10 = 4· (3)9 a10 = 78,732

  6. Geometric Means As we saw with arithmetic means, you are given two terms of a geometric sequence and are asked to find the terms between, these terms between are called geometric means. Find the three geometric means between 3.12 and 49.92. 3.12, _____, _____, _____, 49.92 an = a1 · r n – 1 a5 = 3.12 · r 5 – 1 49.92 = 3.12 r 4 16 = r 4 ±2 = r So… 24.96 –24.96 6.24 – 6.24 12.48 a1 a2 a3 a4 a5

  7. Chapter 11 Sec 4 Geometric Series

  8. Geometric Series Geometric Sequence Geometric Series. 1, 2, 4, 8, 16 1 + 2 + 4 + 8 + 16 4, –12, 36 4 + (–12) + 36 Sn represents the sum of the first n terms of a series. For example, S4 is the sum of the first four terms.

  9. Example 1 Evaluate

  10. Only have the first and last terms? You can use the formula for finding the nth term in (an = a1 · r n – 1) conjunction with the sum formula when you don’t know n. an · r = a1 · r n – 1 · r an · r = a1 · r n

  11. Example 3 Find a1 in a geometric series for which S8 = 39,360and r = 3.

  12. Chapter 11 Sec 5 Infinite Geometric Series

  13. Infinite Geometric Series Any geometric series with an infinite number of terms. Consider the infinite geometric series You have already learned to find the sum Sn of the first n terms, this is called partial sum for an infinite series. Notice that as n increases, the partial sum levels off and approaches a limit of one. This leveling-off behavior is characteristic of infinite geometric series for which | r | < 1.

  14. Sum of an Infinite Series Lets use the formula for the sum of a finite series to find a formula for an infinite series. If –1 < r < 1 , the value if rnwill approach 0 as n increases. Therefore the partial sum of the infinite series will approach

  15. Example 1 Find the sum of each infinite geometric series, if it exists. First find the value of r to determine if the sum exists.

  16. Example 2: Sigma Time… Evaluate

  17. Repeeeeating Decimal typo intentional Write 0.39 as a fraction. S = 0.39 S = 0 .393939393939… then 100S = 39.393939393939… Subtract 100S – S – S = 0 .393939393939… 99S = 39

  18. Daily Assignment • Chapter 11 Sections 3 – 5 • Study Guide (SG) • Pg 145 – 150 Odd

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