1. Arithmetic Sequences (9.2)
Common difference

2. SAT Prep
Quick poll!

3. Start with calculator review of sequences and partial sums
There are three handy hand-outs:
Sequences on the TI-84/84 Plus
Partial sums on the TI 84/ 84 Plus
Sequence mode on the TI 84/ 84 Plus
Let’s see what sort of info they contain.

4. A sequence next Give the first five terms of the sequence for a1 = 2
an+1 = an + 5
What is the pattern for the terms?
What is a possible explicit formula?

5. A sequence next Give the first five terms of the sequence for
an+1 = an + 5
What is the pattern for the terms?
A common difference of 5.

6. Arithmetic sequences
If the pattern between terms in a sequence is a common difference, the sequence is
arithmetic. The difference is d.
Recursive: a1
an+1 = an + d So, d = an+1 - an

7. Arithmetic sequences
If the pattern between terms in a sequence is a common difference, the sequence is
arithmetic.
Explicit:
an = a1 + (n-1) d
(In other words, find the nth term by adding (n-1) d’s to the first term.)

8. Use it Give the first five terms for the sequence an = 2 + (n-1) 9
Write the recursive formula.

9. Use it Give the first five terms for the sequence an = 2 + (n-1) 9
2, 11, 20, 29, 38
Write the recursive formula.
a1 = 2
an+1 = an + 9

10. Use it
an = 2 + (n-1) 9
Find the 10th term of the sequence.

11. Use it an = 2 + (n-1) 9 Find the 10th term of the sequence.
= 2 + (9)9 =
= 83

12. Use it
If the fourth term of a sequence is 5 and the ninth term is 20, find the sixth term.
The key is to find the values for the first term and d.
Start by writing the equations:
20 = a1 + (9 - 1) d 20 = a1 + 8d
5 = a1 + (4 - 1) d 5 = a1 + 3d

13. Use it
If the fourth term of a sequence is 5 and the ninth term is 20, find the sixth term.
Next, use a system of equations to solve for d:
20 = a1 + 8d
-(5 = a1 + 3d)
15 = 5d
3 = d

14. Use it
If the fourth term of a sequence is 5 and the ninth term is 20, find the sixth term.
If d = 3, we can find the first term:
5 = a1 + (4 - 1) 3
5 = a1 + 9
a1 = -4

15. Use it
If the fourth term of a sequence is 5 and the ninth term is 20, find the sixth term.
If d = 3, and a1 = -4 we can find the explicit equation:
an = -4 + (n-1) 3
And the sixth term is a6 = -4 + (5)3 = 11.

16. Partial sums
Add the first 10 terms of our first sequence.

17. Partial sums Add the first 10 terms of our first sequence.
How long did that take? Want a short cut?

18. Partial sums Add the first 10 terms of our first sequence.
…+49 = 10(49).
We want only half of this, so the sum is 5 (49), or 5(first term + last term), which is 245.

19. Partial sums In general, the partial sum for an arithmetic sequence is
Sn = n/2 (a1 + an)
or, if we substitute our explicit formula for an:
Sn = n/2 (2a1 + (n - 1) d)

20. Partial sums In general, the partial sum for an arithmetic sequence is
Sn = n/2 (a1 + an)
Sn = n/2 (2a1 + (n - 1) d)
Test it with our sequence: n = 10, a1 = 2, a10 = 47
S10 = 10/2(2 + 47) = 5(49) = 245

21. Partial sums
Find the sum of the first 100 positive integers.

22. Partial sums Find the sum of the first 100 positive integers.
This problem was posed to Karl Friedrich Gauss ( ) in third grade, and he determined the pattern.
S100 = 100/2(1+100) = 50(101) = 5050

23. Partial sums Find the sum of the first 50 positive even integers.
How does this compare to the sum of the first 100 positive integers?

24. Partial sums Find the sum of the first 50 positive even integers.
How does this compare to the sum of the first 100 positive integers? Let’s look at it in summation notation.
S50 = 50/2( ) = 25(102) = 2550