Warm Up: Find the following sums
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Warm Up: Find the following sums 1. 2. 3. 4.
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Warm Up: Find the following sums. 1. . 2. . 3. . 4. . Sequences. Sometimes it is relatively easy to find the pattern of a sequence. and sometimes it is a little more difficult. Finding Successive Differences. 74. 14 22 32 44 58 ___. 8 10 12 14 . - PowerPoint PPT Presentation
Transcript of Warm Up: Find the following sums
Warm Up: Find the following sums1.
2.
3.
4.
Sequences
Sometimes it is relatively easy to find the pattern of a sequence13,18,23,28,33,___and sometimes it is a little more difficult
2,6,22,56,114,___
Finding Successive Differences
14 22 32 44 58 ___14,22,32,44,58,___
8 10 12 14
2 2 2 216
74
Finding Successive Differences
2 6 22 56 144 ___2,6,22,56,114,___
4 16 34 58
12 18 246 6 6
3088
202
Find the next term in the sequence
5 15 37 77 141 ___
5,15,37,77,141,___
10 22 40 64
12 18 246 6 6
3094
235
Sums of NumbersRecall that we were able to quickly find the sum of consecutive integers by using the Gauss method1+2+3+…+99+100 =
(100 ÷ 2)(101) =5050
Sums of NumbersWrite a formula for finding the sum of the first n counting numbers
1+2+3+…+(n-2)+(n-1)+n=(n ÷ 2)(n+1) =
Sums of NumbersHere 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) n n+1S = 2
Sums of Odd Counting NumbersSee 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
Sums of Odd Counting Numbers1=11+3=
41+3+5=91+3+5+7
=25 2S =n
SummarySum of n
consecutive numbers
2S =n
Sum of n consecutive
odd numbers n n+1S = 2
Find the sum of the first n consecutive even numbers
S=2+4+6+8+…+2nS=1+2+3+…+(n-2)+
(n-1)+n2S=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
S=2S S =n n+12
n n+12=S S = 2 ×2
S =n n+1
Figurate NumbersNumbers and sequences of numbers that are formed based on the geometric arrangement of points
Figurate Number Examples
2 3 4 5 1 1 1
How many dots are in the next figure?
16 21
Figure 1 2 3 4 5 … n# of Dots 1 3 6 10 15 … ?
How many dots are in the 99th figure?
# of dots1
1 1
# of dots136
Figure1234
1 2 31 2 3 4
1 2 3 4510
5 51n
The number of dots in the nth figure is the same as the sum of the first n counting numbers
HomeworkSequence, Sums and Figurative
Numbers Worksheet
