section 11 2 arithmetic sequences
Download
Skip this Video
Download Presentation
Section 11.2 Arithmetic Sequences

Loading in 2 Seconds...

play fullscreen
1 / 16

Section 11.2 Arithmetic Sequences - PowerPoint PPT Presentation


  • 224 Views
  • Uploaded on

Section 11.2 Arithmetic Sequences. Objectives: Recognizing arithmetic sequences by their formulas Finding the first term, the common difference, and find partial sums. Definition of Arithmetic Sequence.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Section 11.2 Arithmetic Sequences' - jolie-farley


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
section 11 2 arithmetic sequences

Section 11.2 Arithmetic Sequences

Objectives:

Recognizing arithmetic sequences by their formulas

Finding the first term, the common difference, and find partial sums.

definition of arithmetic sequence
Definition of Arithmetic Sequence

An arithmetic sequence is in which the difference between each term and the preceding term is always constant.

For example: 2,4,6,8,… is an arithmetic sequence because the difference between each term is 2.

1,4,9,16,25,… is not arithmetic because the difference between each term is not the same.

ex 1 are the following sequences arithmetic if so what is the common difference
Ex.1 Are the following sequences arithmetic? If so, what is the common difference.
  • 14, 10, 6, 2, -2, -6, -10,…

Yes, the sequence is arithmetic because the difference between each term is constant. The common difference is -4

  • 3, 5, 8, 12, 17, …

No, the sequence is not arithmetic because the difference between each term in not the same.

formula for n th term of arithmetic sequence
Formula for nth term of arithmetic sequence:

The formula for the nth term of an arithmetic sequence is

an =a1 + (n – 1)d

ex 2 if a n is an arithmetic sequence with a 1 3 and a 2 4 5 then
Ex 2. If an is an arithmetic sequence with a1= 3 and a2 = 4.5 then

a) Find the first 5 terms of the sequence

The first 5 terms are 3, 4.5, 6, 7.5, and 9

b) Find the common difference, d.

d = 1.5

ex3 find the first six terms and the 300 th term of the arithmetic sequence 13 7
Ex3. Find the first six terms and the 300th term of the arithmetic sequence 13, 7,…

Since a1=13 and a2=7 then the common difference is 7 - 13 = -6.

So the nth term is

an =a1 + (n – 1)d

an = 13 – 6(n – 1)

slide7
From that, we find the first six terms:

13, 7, 1, –5, –11, –17, . . .

  • The 300th term is:

a300 = 13 – 6(299) = –1781

ex 4 the 11th term of an arithmetic sequence is 52 and the 19 th term is 92 find the 1000 th term
Ex 4. The 11th term of an arithmetic sequence is 52, and the 19th term is 92. Find the 1000th term.

To find the nth term of the sequence, we need to find a and d in the formula an= a + (n – 1)d

From the formula, we get: a11 = a + (11 – 1)d =a + 10d a19 = a + (19 – 1)d =a + 18d

slide9
Since a11 = 52 and a19 = 92, we get the two equations:
    • Solving this for a and d, we get: a = 2, d = 5
    • The nth term is: an = 2 + 5(n – 1)
    • The 1000th term is: a1000 = 2 + 5(999) = 4997
partial sums of arithmetic sequences
Partial Sums of Arithmetic Sequences

For the arithmetic sequence an =a + (n – 1)d, the nth partial sum is given by either of these formulas.

ex 5 find the sum of the first 40 terms of the arithmetic sequence 3 7 11 15
Ex 5. Find the sum of the first 40 terms of the arithmetic sequence 3, 7, 11, 15, . . .
  • Here, a = 3 and d = 4.
  • Using Formula 1 for the partial sum of an arithmetic sequence, we get: S40 = (40/2) [2(3) + (40 – 1)4] = 20(6 + 156) = 3240
ex 6 find the sum of the first 50 odd numbers
Ex 6. Find the sum of the first 50 odd numbers.
  • The odd numbers form an arithmetic sequence with a = 1 and d = 2.
  • The nth term is: an = 1 + 2(n – 1) = 2n – 1
  • So, the 50th odd number is: a50 = 2(50) – 1 = 99
slide14
Ex 7. An amphitheater has 50 rows of seats with 30 seats in the first row, 32 in the second, 34 in the third, and so on. Find the total number of seats.
  • The numbers of seats in the rows form an arithmetic sequence with a = 30 and d = 2.
    • Since there are 50 rows, the total number of seats is the sum
    • The amphitheater has 3950 seats.
ex 8 how many terms of the arithmetic sequence 5 7 9 must be added to get 572
Ex 8 How many terms of the arithmetic sequence 5, 7, 9, . . . must be added to get 572?

We are asked to find n when Sn = 572.

  • Substituting a = 5, d = 2, and Sn = 572 in Formula 1, we get:
    • This gives n = 22 or n = –26.
    • However, since n is the number of terms in this partial sum, we must have n = 22.
ad