 # Section 5 7 Arithmetic and Geometric Sequences Copyright

• Slides: 19 Section 5. 7 Arithmetic and Geometric Sequences Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Arithmetic Sequences Geometric Sequences 5. 7 -2 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Sequences A sequence is a list of numbers that are related to each other by a rule. The terms are the numbers that form the sequence. 5. 7 -3 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Arithmetic Sequence An arithmetic sequence is a sequence in which each term after the first term differs from the preceding term by a constant amount. The common difference, d, is the amount by which each pair of successive terms differs. To find the difference, simply subtract any term from the term that directly follows it. 5. 7 -4 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: An Arithmetic Sequence with a Negative Difference Write the first five terms of the arithmetic sequence with first term 9 and a common difference of – 4. Solution The first five terms of the sequence are 9, 5, 1, – 3, – 7 5. 7 -5 Copyright 2013, 2010, 2007, Pearson, Education, Inc. General or nth Term of an Arithmetic Sequence For an arithmetic sequence with first term a 1 and common difference d, the general or nth term can be found using the following formula. an = a 1 + (n – 1)d 5. 7 -6 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Determining the 12 th Term of an Arithmetic Sequence Determine the twelfth term of the arithmetic sequence whose first term is – 5 and whose common difference is 3. Solution Replace: a 1 = – 5, n = 12, d = 3 an = a 1 + (n – 1)d a 12 = – 5 + (12 – 1)3 = – 5 + (11)3 = 28 5. 7 -7 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: Determining an Expression for the nth Term Write an expression for the general or nth term, an, for the sequence 1, 6, 11, 16, … Solution Substitute: a 1 = 1, d = 5 an = a 1 + (n – 1)d = 1 + (n – 1)5 = 1 + 5 n – 5 = 5 n – 4 5. 7 -8 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Sum of the First n Terms of an Arithmetic Sequence The sum of the first n terms of an arithmetic sequence can be found with the following formula where a 1 represents the first term and an represents the nth term. 5. 7 -9 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 5: Determining the Sum of an Arithmetic Sequence Determine the sum of the first 25 even natural numbers. Solution The sequence is 2, 4, 6, 8, 10, …, 50 Substitute a 1 = 2, a 25 = 50, n = 25 into the formula 5. 7 -10 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 5: Determining the Sum of an Arithmetic Sequence Solution a 1 = 2, a 25 = 50, n = 25 5. 7 -11 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Geometric Sequences A geometric sequence is one in which the ratio of any term to the term that directly precedes it is a constant. This constant is called the common ratio, r. r can be found by taking any term except the first and dividing it by the preceding term. 5. 7 -12 Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 5 and whose common ratio, r, is 2. Solution The first five terms of the sequence are 5, 10, 20, 40, 80 5. 7 -13 Copyright 2013, 2010, 2007, Pearson, Education, Inc. General or nth Term of a Geometric Sequence For a geometric sequence with first term a 1 and common ratio r, the general or nth term can be found using the following formula. an = a 1 r n– 1 5. 7 -14 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 7: Determining the 12 th Term of a Geometric Sequence Determine the twelfth term of the geometric sequence whose first term is – 4 and whose common ratio is 2. Solution Replace: a 1 = – 4, n = 12, r = 2 an = a 1 r n– 1 a 12 = – 4 • 212– 1 = – 4 • 211 = – 4 • 2048 = – 8192 5. 7 -15 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 8: Determining an Expression for the nth Term Write an expression for the general or nth term, an, for the sequence 2, 6, 18, 54, … Solution Substitute: a 1 = 2, r = 3 an = a 1 r n– 1 = 2(3)n– 1 5. 7 -16 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Sum of the First n Terms of an Geometric Sequence The sum of the first n terms of an geometric sequence can be found with the following formula where a 1 represents the first term and r represents the common ratio. 5. 7 -17 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 9: Determining the Sum of an Geometric Sequence Determine the sum of the first five terms in the geometric sequence whose first term is 4 and whose common ratio is 2. Solution Substitute a 1 = 4, r = 2, n = 5 into 5. 7 -18 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 5: Determining the Sum of an Arithmetic Sequence Solution a 1 = 2, r = 2, n = 5 5. 7 -19 Copyright 2013, 2010, 2007, Pearson, Education, Inc.