§ 11.2 Arithmetic Sequences

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 11.2 Arithmetic Sequences Annual U.S. Senator Salaries from 2000 to 2005

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 11.2 Arithmetic Sequences The sequence of annual salaries for U.S. Senators represented by the bar chart is: 142,146,150,154,158,162,… For each term after the first, the difference between that term and the preceding term is constant, namely 4. This sequence is an example of an arithmetic sequence. In an arithmetic sequence, each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the common difference of the sequence.

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 11.2 Arithmetic Sequences Definition of an Arithmetic Sequence An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the common difference of the sequence. In other words, an arithmetic sequence is a linear function whose domain is the set of positive integers.

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 11.2 Arithmetic SequencesEXAMPLE SOLUTION Write the first six terms of an arithmetic sequence with first term -60 and common difference 5. To find the second term, we add 5 to the first term, -60, giving -55. For the next term, we add 5 to -55, and so on. The first six terms are -60, -55, -50, -45, -40, and -35.

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 11.2 Arithmetic Sequences General Term of an Arithmetic Sequence The nth term (the general term) of an arithmetic sequence with first term and common difference d is

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 11.2 Arithmetic SequencesEXAMPLE SOLUTION Find the 200 th term of the arithmetic sequence whose first term is -40 and whose common difference is 5. To find the 200 th term,, we replace n in the formula with 200, with -40. The 200 th term is 955.

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 11.2 Arithmetic Sequences in ApplicationEXAMPLE SOLUTION Company A pays $24,000 yearly with raises of $1600 per year. Company B pays $28,000 yearly with raises of $1000 per year. Which company will pay more in year 10? How much more? Since both companies increase the salaries of their employees by a constant amount each year (common difference), their salary schedules are arithmetic sequences. Therefore, we will use the formulas given in this section to determine how much each company pays in year 10. That is, we will determine for both companies and then compare the results. Since the initial salaries for the two companies are $24,000 and $28,000, these will be the values for for the respective companies.

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 11.2 Arithmetic SequencesCONTINUED Company A Company B Therefore, in year 10, Company A will pay more. In that year, Company A will pay $38,400 - $37,000 = $1400 more.

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 11.2 Arithmetic Sequences The Sum of the First n Terms of an Arithmetic Sequence The sum,, of the first n terms of an arithmetic sequence is given by in which is the first term and is the nth term.

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 11.2 Arithmetic SequencesEXAMPLE SOLUTION Find the sum of the first 50 terms of the arithmetic sequence: By find the sum of the first 50 terms of the sequence, we are finding the sum of the first 50 terms of the arithmetic sequence having and d = 6. To find the sum of the first 50 terms,, we replace n in the formula with , -9, -3, 3,...

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 11.2 Arithmetic Sequences We use the formula for the general term of a sequence to find. The common difference, d, of -15, -9, -3, 3,..., is 6. CONTINUED This is the formula for the nth term of an arithmetic sequence. Use it to find the 50 th term. Substitute 50 for n, 6 for d, and -15 for. Now we are ready to find the sum of the first 50 terms of -15, -9, -3,

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 11.2 Arithmetic SequencesCONTINUED Use the formula for the sum of the first n terms of an arithmetic sequence. Let n = 50,, and The sum of the first 50 terms of the given sequence is Equivalently, the 50 th partial sum of the sequence is 6600.

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 11.2 Arithmetic SequencesEXAMPLE SOLUTION Find the following sum: By evaluating the first three terms and the last term, we see that ; d, the common difference, is 2 – 4 = -2; and, the last term, is -74.

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 11.2 Arithmetic Sequences Use the formula for the sum of the first n terms of an arithmetic sequence. Let n = 40,, and CONTINUED Thus,

Blitzer, Intermediate Algebra, 5e – Slide #16 Section 11.2 Arithmetic Sequences in ApplicationEXAMPLE SOLUTION A theater has 30 seats in the first row, 32 seats in the second row, increasing by 2 seats each row for a total of 26 rows. How many seats are there in the theater? Each row of seats has two more seats than the row in front of it. That is, there is a common difference between each row and the row that immediately precedes it. Therefore, the number of seats in each row generates an arithmetic sequence. Therefore, we will use the formulas given in this section to determine how many seats there are in the theater. Since each row has 2 more seats than the row before it, d = 2. Since the first row has 30 seats, Since there are 26 rows, n = 26.

Blitzer, Intermediate Algebra, 5e – Slide #17 Section 11.2 Arithmetic Sequences in Application So we now need only determine and then we can find the sum of all the seats. That is, we will then be able to determine CONTINUED This is the formula for the nth term of an arithmetic sequence. Use it to find the 26 th term. Substitute 26 for n, 2 for d, and 30 for. Now we are ready to find the sum of the finite sequence.

Blitzer, Intermediate Algebra, 5e – Slide #18 Section 11.2 Arithmetic Sequences in ApplicationCONTINUED Therefore, there are 1430 seats in the theater. Use the formula for the sum of the first n terms of an arithmetic sequence. Let n = 26,, and