NUMBER PATTERNS ARITHMETIC SEQUENCE
Number patterns are the patterns in a list number that -every term after the first term is obtained by adding a fixed
follows a certain sequence. number
Common difference- the difference bet. consecutive terms
APPLICATIONS:
𝒅 = 𝒂𝟐 − 𝒂𝟏
1. Financial Planning- budgeting Formula of arithmetic sequence: 𝒂𝒏 = 𝒂 + (𝒏 − 𝟏)𝒅
2. Prediction Analysis- weather forecasting, stock 12, 7, 2, … 16th term
market predictions, sales forecasting a=12 d= -5
3. Engineering and Constructions- optimize designs and 𝑎𝑛= 12 + (𝑛 − 1) − 5
𝑎16 = 12 + (16 − 1) − 5
algorithms, ensure structural integrity
𝑎16 = 12 + (15) − 5
4. Computer science and Coding- software 𝑎16 = 12 + (−75)
development, optimizing algorithms, creating 𝑎16 = −63
efficient code
5. Data Analysis and Statistics- identifying trends, Suppose you have a dataset where the first data point is 20,
interpreting and analyzing data and each subsequent data point decreases by 4. What is the
100th data point in the sequence?
Pattern- a term used to describe repeating objects/events.
a= 20 d= -4
Rule- operations and fixed numbers involved in a sequence. 𝑎𝑛 = 20 + (𝑛 − 1) − 4
𝑎100 = 20 + (100 − 1) − 4
4, 25, 46, 67, …. Add 21 𝑎100 = 20 + (99) − 4 The 100th data point in the
𝑎100 = 20 + (−396) sequence is -376.
Sequence- a list of items/objects that has a pattern 𝑎100 = −376
Series- sum of all the terms in a sequence
1. 1 + 2 + 3 + 4 + …. + 15 GEOMETRIC SEQUENCE
x= 1 + 2 + 3 + 4 +…+15 -each term, after the first, is obtained by multiplying the
x= 15+14+13+12…+1 preceding term by a common ratio.
-the product of consecutive terms should be constant to be
2x= 16+16+16+16…+16 considered geometric
2x=16(15) Common ratio- dividing any term, except the first, by the
𝒂
preceding term. 𝒓 = 𝟐
2𝑥 240 𝒂𝟏
= Formula of geometric sequence: 𝒂𝒏 = 𝒂 (𝒓)𝒏−𝟏
2 2
4, -12, 36, -108, …. 8th term
x= 120
a= 4 r= -3
2. what is the max. no. of slices if we cut the cake 100 times? 𝑎𝑛 = 4 (−3)𝑛−1
𝑎8 = 4 (−3)8−1
No. of cuts No. of slices Pattern
𝑎8 = 4 (−3)7
0 1 1 𝑎8 = 4 (−2,187)
𝑎8 = −8,748
1 2 1+1= 2
2 4 1+1+2= 4 Jennifer started a business and invested $10,000 into it. She
3 7 1+1+2+3= 7 expects the business to grow at a rate of 5% annually. How
much money will Jennifer’s business be worth after 5 years?
4 11 1+1+2+3+4= 11 a= 10,000 r= 1.05
100 5051 1+1+2+3+4+…+100 𝑎𝑛 = 10,000 (1.05)𝑛−1
𝑎5 = 10,000 (1.05)5−1 Jennifer’s business after 5 years
x= 1 + 2 + 3 + 4 +…+100 x= 5,050 + 1 𝑎5 = 10,000 (1.05)4 will be worth $12, 115.
x= 100+99+98+97+…+1 x= 5,051 𝑎5 = 10,000 (1.2155)
𝑎5 = 12,155
2x= 101+101+101+…101 The maximum no. of
2x=101(100) slices if we cut the cake
100 times is 5,051. FIBONACCI SEQUENCE
2𝑥 10,000
= -a Fibonacci sequence is infinite and each term, starting from
2 2
the third, is the sum of the preceding term two Fibonacci
terms.
-a sequence whose terms are: 1,1,2,3,5,8,13,21,34,55…
Number patterns are the patterns in a list number that -every term after the first term is obtained by adding a fixed
follows a certain sequence. number
Common difference- the difference bet. consecutive terms
APPLICATIONS:
𝒅 = 𝒂𝟐 − 𝒂𝟏
1. Financial Planning- budgeting Formula of arithmetic sequence: 𝒂𝒏 = 𝒂 + (𝒏 − 𝟏)𝒅
2. Prediction Analysis- weather forecasting, stock 12, 7, 2, … 16th term
market predictions, sales forecasting a=12 d= -5
3. Engineering and Constructions- optimize designs and 𝑎𝑛= 12 + (𝑛 − 1) − 5
𝑎16 = 12 + (16 − 1) − 5
algorithms, ensure structural integrity
𝑎16 = 12 + (15) − 5
4. Computer science and Coding- software 𝑎16 = 12 + (−75)
development, optimizing algorithms, creating 𝑎16 = −63
efficient code
5. Data Analysis and Statistics- identifying trends, Suppose you have a dataset where the first data point is 20,
interpreting and analyzing data and each subsequent data point decreases by 4. What is the
100th data point in the sequence?
Pattern- a term used to describe repeating objects/events.
a= 20 d= -4
Rule- operations and fixed numbers involved in a sequence. 𝑎𝑛 = 20 + (𝑛 − 1) − 4
𝑎100 = 20 + (100 − 1) − 4
4, 25, 46, 67, …. Add 21 𝑎100 = 20 + (99) − 4 The 100th data point in the
𝑎100 = 20 + (−396) sequence is -376.
Sequence- a list of items/objects that has a pattern 𝑎100 = −376
Series- sum of all the terms in a sequence
1. 1 + 2 + 3 + 4 + …. + 15 GEOMETRIC SEQUENCE
x= 1 + 2 + 3 + 4 +…+15 -each term, after the first, is obtained by multiplying the
x= 15+14+13+12…+1 preceding term by a common ratio.
-the product of consecutive terms should be constant to be
2x= 16+16+16+16…+16 considered geometric
2x=16(15) Common ratio- dividing any term, except the first, by the
𝒂
preceding term. 𝒓 = 𝟐
2𝑥 240 𝒂𝟏
= Formula of geometric sequence: 𝒂𝒏 = 𝒂 (𝒓)𝒏−𝟏
2 2
4, -12, 36, -108, …. 8th term
x= 120
a= 4 r= -3
2. what is the max. no. of slices if we cut the cake 100 times? 𝑎𝑛 = 4 (−3)𝑛−1
𝑎8 = 4 (−3)8−1
No. of cuts No. of slices Pattern
𝑎8 = 4 (−3)7
0 1 1 𝑎8 = 4 (−2,187)
𝑎8 = −8,748
1 2 1+1= 2
2 4 1+1+2= 4 Jennifer started a business and invested $10,000 into it. She
3 7 1+1+2+3= 7 expects the business to grow at a rate of 5% annually. How
much money will Jennifer’s business be worth after 5 years?
4 11 1+1+2+3+4= 11 a= 10,000 r= 1.05
100 5051 1+1+2+3+4+…+100 𝑎𝑛 = 10,000 (1.05)𝑛−1
𝑎5 = 10,000 (1.05)5−1 Jennifer’s business after 5 years
x= 1 + 2 + 3 + 4 +…+100 x= 5,050 + 1 𝑎5 = 10,000 (1.05)4 will be worth $12, 115.
x= 100+99+98+97+…+1 x= 5,051 𝑎5 = 10,000 (1.2155)
𝑎5 = 12,155
2x= 101+101+101+…101 The maximum no. of
2x=101(100) slices if we cut the cake
100 times is 5,051. FIBONACCI SEQUENCE
2𝑥 10,000
= -a Fibonacci sequence is infinite and each term, starting from
2 2
the third, is the sum of the preceding term two Fibonacci
terms.
-a sequence whose terms are: 1,1,2,3,5,8,13,21,34,55…
