5Arithmetic Sequences A sequence whose consecutive terms have a common difference is called an arithmetic sequence.
6Example 1 – Examples of Arithmetic Sequences a. The sequence whose n th term is 4n + 3 is arithmetic.For this sequence, the common difference between consecutive terms is 4.7, 11, 15, 19, , 4n + 3, . . .b. The sequence whose nth term is 7 – 5n is arithmetic.For this sequence, the common difference between consecutive terms is – 5.2, –3, – 8, –13, , 7 – 5n, . . .Begin with n = 1.11 – 7 = 4Begin with n = 1.–3 – 2 = –5
7Example 1 – Examples of Arithmetic Sequences cont’dc. The sequence whose nth term is is arithmetic.For this sequence, the common difference between consecutive terms isBegin with n = 1.
8Arithmetic SequencesThe sequence 1, 4, 9, 16, , whose n th term is n2, is not arithmetic. The difference between the first two terms isa2 – a1 = 4 – 1 = 3but the difference between the second and third terms isa3 – a2 = 9 – 4 = 5.
9Example 2 – Finding the nth Term of an Arithmetic Sequence Find a formula for the n th term of the arithmetic sequence whose common difference is 3 and whose first term is 2.Solution:You know that the formula for the n th term is of the forman = a1 + ( n – 1)d.Moreover, because the common difference is d = 3 and the first term is a1 = 2, the formula must have the forman = 2 + 3(n – 1).Substitute 2 for a 1 and 3 for d.
10Example 2 – Solution So, the formula for the n th term is an = 3n – 1. cont’dSo, the formula for the n th term isan = 3n – 1.The sequence therefore has the following form.2, 5, 8, 11, 14, , 3n – 1, . . .
11Arithmetic SequencesIf you know the n th term of an arithmetic sequence and you know the common difference of the sequence, you can find the (n + 1)th term by using the recursion formulaan + 1 = an + d.With this formula, you can find any term of an arithmetic sequence, provided that you know the preceding term.For instance, if you know the first term, you can find the second term. Then, knowing the second term, you can find the third term, and so on.Recursion formula
13The Sum of a Finite Arithmetic Sequence There is a simple formula for the sum of a finite arithmetic sequence.
14Example 5 – Finding the Sum of a Finite Arithmetic Sequence Find the sum:Solution:To begin, notice that the sequence is arithmetic (with a common difference of 2).Moreover, the sequence has 10 terms. So, the sum of the sequence isSn = (a1 + an)Formula for the sum of an arithmetic sequence
15Example 5 – Solution = (1 + 19) = 5(20) = 100. cont’d = (1 + 19)= 5(20)= 100.Substitute 10 for n, 1 for a1, and 19 for an.Simplify.
16The Sum of a Finite Arithmetic Sequence The sum of the first n terms of an infinite sequence is the n th partial sum.The n th partial sum can be found by using the formula for the sum of a finite arithmetic sequence.
18Example 8 – Prize MoneyIn a golf tournament, the 16 golfers with the lowest scoreswin cash prizes. First place receives a cash prize of $1000,second place receives $950, third place receives $900, andso on. What is the total amount of prize money?Solution:The cash prizes awarded form an arithmetic sequence in which the first term is a1 = 1000 and the common difference is d = – 50.
19Example 8 – Solution Because an = 1000 + (– 50)(n – 1) cont’dBecausean = (– 50)(n – 1)you can determine that the formula for the n th term of the sequence is an = – 50nSo, the 16th term of the sequence isa16 = – 50(16)and the total amount of prize money isS16 == 250,
20Example 8 – Solution S16 = (a1 + a16) = (1000 + 250) = 8(1250) cont’dS16 = (a1 + a16)= ( )= 8(1250)= $10,000.n th partial sum formulaSubstitute 16 for n, 1000 for a1, and 250 for a16.Simplify.