2. Alternating signs means we have (-1)n-1 power.
Make a T-chart and find a pattern with the difference tests.
1st level Difference Test.
Linear equation with a slope of 2.
What needs to be done to 2, 4, 6, 8 to get 1, 3, 5, 7?
What needs to be done to 1, 4, 9, 16 to get 2, 5, 10, 17?
2nd level Difference Test. Quadratic equation

3. Do you see a pattern?

6. The k values are 1 bigger than the powers on the 2.

7. Notice that all the properties creates summations that all start at k = 1 and ends with n.

8. n = n = Find each sum. Property #2 and #6. Property #1. Property #3.

9. Property #2, #3, and #4.
n =
Property #7, #6, and #1.

10. Property #2.
Property #5.
n =
Not k = 1, Property #5

11. 1st level difference test.
a40 = 2 + 4(40 – 1)
n = 40
a1 = 2 +4 +4 +4 +4
d = 4
a40 = 2 + 4(39) = 158
Treat the terms like points, as we did in Sect. 9.1, ( 4, 20 ) and ( 13, 65 ).
The difference, d, is the same as the slope. Find the slope between the 2 points.
Find a1, use ( 4, 20 ) and ( 1, a1 ) with the slope formula.

12. This is a visual approach to solving this problem
This is a visual approach to solving this problem. Make consecutive blanks for each term. a1
____,
____,
____,
____,
____,
____,
____,
____,
____,
____,
____,
____,
____
Make an equation from 20 to 65 with the 9 d’s.
Now subtract the d value backwards for the 1st term.

13. 101 n a1 a150 = ? a1 an d = -50 n = ? 101 times how many groups?
101 ( 50 ) = 5050
101
Build the formula. Where did the numbers come from?
These two formulas go hand in hand. n a1
d = 11
a150 = ?
+11
+11
+11
+11 a1 an
d = -50
n = ?
Solve for n.
Find these values.

14. Make an equation from 36 to 16 with the 2 r’s.*3 *3 *3
Make an equation from 36 to 16 with the 2 r’s.
____,
____,
____,
____,
____
With two r values gives us 2 equations.

15. n = 10 a1 = 5 r = 2 Multiply both sides by a –r.
This creates opposite terms.
Add the equations together and cancel terms.
Factor out the S.
Factor out the a1
Divide by (1 – r).
n = 10
a1 = 5
r = 2

16. ? ? ( ½ )oo will continue to get smaller until it is zero!
? ?
( ½ )oo will continue to get smaller until it is zero!
Dividing by ( ½ ) is the same as multiplying by the reciprocal of 2.

17. Determine when a series diverge or converge.
If a finite sum approaches a number L as , we say the
infinite geometric series converges. A series
diverges when the sum is
An infinite geometric series will always converge when
This gives us two formulas for geometric series.
An infinite geometric series will always diverge when

18. 2nd STAT – OPS #5 is the sequence formula.
seq(function,variable,starting value,ending value)
Find the first 5 terms of
2nd STAT – MATH #5 is the sum formula.
Used for Series questions.
sum(seq(function,variable,starting value,ending value))
Find