6.1 Sequences and Arithmetic Sequences

3/20/2013

Sequence

a list of terms with a particular order.

Ex. 2, 5, 8, 11, 14,… (increasing by 3)

or 2, 5, 10, 17, 26,… (increasing by 3, 5, 7, 9, etc)

If the terms of a sequence have a recognizable pattern, you may be able to write a rule for the nth term of the sequence.

Find the pattern by writing an expression (or rule) for the nth term:1.) Each term increases by 4,1st term: 32nd term: 3 +4(1) = 73rd term: 3 + 4(2) = 114th term: 3 + 4(3) = 15Nth term: 3 + 4 (n-1)which simplifies to

Or

1.) If you notice, each term is 1 less than multiples of 4 (4, 8, 12, 15, 20).Therefore nth term = 4n -1

Find the pattern by writing an expression (or rule) for the nth term:2.) Hint: Think perfect squares.Each term is 1 less than a perfect square. (1st term) (2nd term) (3rd term) (nth term)

Write the first five terms of the following sequences:

3.)

Without doing the rest, you can see the pattern.Each term increases by 2.7, 9, 11, 13, 15

Write the first five terms of the following sequences:

4.)

Without doing the rest, you can see the pattern.

Arithmetic Sequences

are sequences where the difference between consecutive terms is always the same number. This number is called the common difference (d).

Ex: ,…

Common difference: 4

Determine whether the sequence is arithmetic

10, 8, 6, 4, 2,….

3, 2.5, 2, 1.5, 1,…

Yes

Yes

No

Rule for finding the nth term of arithmetic sequence:

Ex: Each term increases by 4d (common difference) = 41st term: 32nd term: 3 +4(1) = 73rd term: 3 + 4(2) = 114th term: 3 + 4(3) = 15Nth term: 3 + 4 (n-1)

Rule:

= the value of the nth term= the first number in the sequenced = common differencen = nth term.

5.) Find the 20th term of the sequence:,…

The story is told of a grade school teacher In the 1700's that wanted to keep her class busy while she graded papers so she asked them to add up all of the numbers from 1 to 100. These numbers are an arithmetic sequence with common difference 1. Carl Friedrich Gauss was in the class and had the answer in a minute or two (remember no calculators in those days). This is what he did:

1 + 2 + 3 + 4 + 5 + . . . + 96 + 97 + 98 + 99 + 100

sum is 101

sum is 101

With 100 numbers there are 50 pairs that add up to 101. 50(101) = 5050

Sum of a finite Arithmetic Series

Where is first term is the last term and n the number of terms

Find the sum of the arithmetic sequence:

6.) 2+4+6+8+10+12+14+16+18

= 90

Find the sum of the

7.) first 40 positive odd integers1, 3, 5, 7…..39

= 400

HomeworkWorksheet 6.1 odd problems only.

“I tried to catch some fog. I mist!”