1. Warm Up: Find the following sums

2. Sequences Sometimes it is relatively easy to find the pattern of a sequence and sometimes it is a little more difficult

3. Finding Successive Differences 14 22 32 44 58 ___ 8 10 12 14 2 2 2 2 16 74

4. Finding Successive Differences 2 6 22 56 144 ___ 4 16 34 58 12 18 24 6 6 30 88 202

5. Find the next term in the sequence 5 15 37 77 141 ___ 10 22 40 64 12 18 24 6 6 30 94 235

6. Sums of Numbers Recall that we were able to quickly find the sum of consecutive integers by using the Gauss method 1+2+3+…+99+100 = (100 ÷ 2)(101) = 5050

7. Sums of Numbers Write a formula for finding the sum of the first n counting numbers 1+2+3+…+(n-2)+(n-1)+n= (n ÷ 2)(n+1) =

8. Sums of Numbers Here is an interesting way to come up with this formula Let S=1+ 2 + 3 +…+(n-1)+n 2S=(n+1)+(n+1)+(n+1)+…+(n+1) + S=n+(n-1)+(n-2)+…+ 2 +1 2S=n(n+1)

9. Sums of Odd Counting Numbers See if you can find the sum of odd consecutive counting numbers. Find the sum of the first n odd counting numbers S=1+3+5+7+…+(2n-1) Hint: Try taking a couple of sample sums

10. Sums of Odd Counting Numbers 1=1 1+3=4 1+3+5=9 1+3+5+7=25

11. Summary Sum of n consecutive numbers Sum of n consecutive odd numbers

12. Find the sum of the first n consecutive even numbers S=2+4+6+8+…+2n S=1+2+3+…+(n-2)+(n-1)+n 2S=2(1+2+3+…+(n-2)+(n-1)+n) 2S=2(1)+2(2)+2(3)+…+2(n) 2S=2+4+6+8+…+2n S=2S

14. Figurate Numbers Numbers and sequences of numbers that are formed based on the geometric arrangement of points

15. Figurate Number Examples

16. 2 3 4 5 1 1 1 How many dots are in the next figure? 1 6 21

17. Figure12345 …n # of Dots1361015 … ? How many dots are in the 99th figure?

18. The number of dots in the nth figure is the same as the sum of the first n counting numbers

19. Homework Sequence, Sums and Figurative Numbers Worksheet