278 CHAPTER 5 Number Theory and the Real Number System The following formula can be used to determine the nth term of a geometric sequence. Example 6 The First Five Terms of a Geometric Sequence Write the first five terms of the geometric sequence whose first term, a ,1 is 3 and whose common ratio, r, is 4. Solution The first term is 3. The second term, determined by multiplying the first term by 4, is 3 4, ⋅ or 12. The third term is 12 4, ⋅ or 48. The fourth term is 48 4 ⋅ or 192. The fifth term is 192 4, ⋅ or 768. Thus, the first five terms of the sequence are 3, 12, 48, 192, 768. 7 Now try Exercise 31 General or th n Term of a Geometric Sequence For a geometric sequence with first term a1 and common ratio r, the general or nth term can be determined using the following formula. a a r n n 1 1 = − Example 7 Determine the Eighth Term of a Geometric Sequence Determine the eighth term of the geometric sequence whose first term is 2− and common ratio is 4. Solution In this sequence, a r 2, 4, 1 = − = and n 8. = Substituting the values into the formula, we obtain a a r a 2 4 2 4 2 16,384 32,768 n n 1 1 8 8 1 7 = = − ⋅ = − ⋅ = − ⋅ = − − − As a check, we have listed the first 8 terms of the sequence. − − − − − − − − 2, 8, 32, 128, 512, 2048, 8192, 32,768 7 Now try Exercise 37 Example 8 Determining an Expression for the nth Term Write an expression for the general or nth term, a ,n of the sequence … 2, 6, 18, 54, . Solution In this sequence, the first term is 2, so a 2. 1 = Note that 3, 6 2 = so r 3. = We substitute a 2 1 = and r 3 = into a a r n n 1 1 = − to obtain an expression for the nth term, a .n a a r 2(3) n n n 1 1 1 = = − − The expression is a 2(3) . n n 1 = − Note that when n a 1, 2(3) 2(1) 2. 1 0 = = = = When n 2, = a 2(3) 6, 2 1 = = and so on. 7 Now try Exercise 43 The following formula can be used to determine the sum of the first n terms of a geometric sequence.
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