9.2 Arithmetic Sequences and Partial Sums

9.2 Arithmetic Sequences

A sequence is arithmetic if the differences betweenconsecutive terms are the same.

That common difference is called d.

For Ex. 7, 11, 15, 19, … , 4n + 3 or2, -3, -8, -13, … , 7 - 5n are arith. seq.’s

Let’s take a look at the first example. What is d, thecommon difference? What is a1?

a1 = 7

a2 = 7 + 4 = 11

a3 = 7 + 2(4) = 15

a4 = 7 + 3(4) = 19

Do you see a pattern?What is a5? What is an?What is the 35th term?

Can you come up with an equation that will give you the nthterm of any arithmetic sequence?

an = a1 + (n - 1)d

Find a formula for the nth term of the arithmetic sequencewhose common difference is 5 and whose 2nd term is 12.What is the 18th term of the sequence?

What do we know? That d = 5 and a2 = 12

a2 = a1 + (2-1)d or a1 + d Substitute in for a2 and dand solve for a1

12 = a1 + 5 and a1 = 7 How now, do we find the 18th term?

a18 = a1 + 17d = 7 + 17(5) = 92

Find the ninth term of the arithmetic sequence whose first two terms are 2 and 9.

What is the common difference, d? d = 7

Now we can find the 9th term because we know a1 and d.Find a9.

a9 = 2 + 8(7) = 58

The fourth term of an arithmetic sequence is 20, and the 13th is 65. Write the first 4 terms of this sequence.

Write the equations of any 4th and 13th terms of anyarithmetic sequence.

a4 = a1 + 3da13 = a1 + 12d

Now fill in what we know and use ellimination to find a1 and d.

So the first four terms are: 5, 10, 15, 20

Homework 1-37 odd

The sum of a finite arithmetic sequence

)(2 1 naan

S +=

Ex. Add the numbers from 1 to 100.

( ) 505010012

100=+=S

Find the sum.

=−∑=

150

1

)611(n

n

It would take way too longto do this by hand. Using the formula just given, we can do it in seconds.

First, we have to find a1 and a150.

a1 =a150 =

Now substitute those into the formula.

675,123)16445(2

150=+=S

Insert 3 arithmetic means between 4 and 15.4 ____ ____ ____ 15

We need to find the common difference.

Write the equation for a5.

a5 = a1+ 4d

15 = 4 + 4d

4

11=d So the sequence is?

15,4

49,

4

38,

4

27,4

An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 in the the second row, 22 in the third row, and so on.How many seats are there in all 20 rows?

So, what do we know? Do we know a1? d?Do we know how many seats are in the lastrow? a20

a20 = 20 + 19(1) = 39 seats in the last row.

Now find S.

)3920(2

20+=S = 590

Homework 39-79 odd