11-3 Geometric Sequences Hubarth Algebra II.

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11-3 Geometric Sequences Hubarth Algebra II

In a geometric sequence, the ratio between consecutive terms is constant. This ratio is called the common ratio. Unlike in an arithmetic sequence, the difference between consecutive terms varies. Ex. 1 Identifying a Geometric Sequences Is the given sequence geometric? If so, identify the common ratio. a. 1, –6, 36, –216, . . . 1, –6, 36, –216 –6 ÷ 1 = –6  –6 36 ÷ –6 = –6  –6 216 ÷ 36 = –6  –6 There is a common ratio of –6. This is a geometric sequence. b. 2, 4, 6, 8, . . . 2, 4, 6, 8 4 ÷ 2 = 2  2  3 2 6 ÷ 4 =  4 3 8 ÷ 6 = There is no common ratio. This is not a geometric sequence.

Ex. 2 Real-World Connection Suppose you have equipment that can enlarge a photo to 120% of its original size. A photo has a length of 10 cm. Find the length of the photo after 5 enlargements at 120%. You need to find the 6th term of the geometric sequence 10, 12, 14.4, . . . an = a1 • r n – 1 Use the explicit formula. a6 = 10 • 1.206 – 1 Substitute a1 = 10, n = 6, and r = 1.20. = 10 • 1.205 Simplify the exponent. 24.883 Use a calculator. After five enlargements of 120%, the photo has a length of about 25 cm.

You can find the geometric mean of any two positive numbers by taking the positive square root of the product of the two numbers 𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑚𝑒𝑎𝑛= 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 Ex. 3 Real-World Connection A family purchased a home for $150,000. Two years later the home was valued at $188,160. If the value of the home is increasing geometrically, how much was the home worth after one year? geometric mean = 150,000 • 188,160 Use the definition. = 28,224,000,000 Multiply. = 168,000 Take the square root.

Practice 1. Write the first 10 terms of this geometric sequence. 5, 15, 45, 135, ……. 5, 15, 45, 135, 405, 1215, 3645, 10,935, 32,805, 98,415 2. Is the sequence 6, -24, 96, -384,…… arithmetic, geometric or neither. Geometric, the common ratio is -4 3. Find the 19th term in each sequence. a. 11, 33, 99, 297,……… b. 20, 17, 14, 11, 8,……. 4, 261, 625, 379 -34 4. Find the missing term of each geometric sequence. It could be the geometric mean or its opposite.. a. 20, , 80 b. 3, , 18.75 c. 28, , 5103 40 or -40 7.5 or -7.5 378 or -378