Patterns
Numerical Patterns A numerical pattern is a list of numbers that follow a predictable rule. Once you determine that rule, you can extend the pattern. The rule must work for every number in the list. think Look at the first number in the list below. What can I do to the first number to get the second number in the list? Look at the second number in the list below. What can I do to the second number to get the third number in the list? 2, 4, 6, 8, 10, 12 Add 2 (2 + 2 = 4) Multiply by 2 (2 × 2 = 4) (4 + 2 = 6) (6 + 2 = 8) (8 + 2 = 10) (10 + 2 = 12)
Once you determine that rule, you can extend the pattern. Geometric Patterns A geometric pattern is a list of geometric shapes that follow a predictable rule. Once you determine that rule, you can extend the pattern. Assigning a letter of the alphabet to each type of shape can help you find the rule. think What comes next in this pattern? The pattern is an ABBB pattern. The next shape in this pattern is a trapezoid. A B
Finding the nth Term ( an ) in an Arithmetic Sequence One type of numerical pattern is an arithmetic sequence. An arithmetic sequence is an ordered set of real numbers. Each number in a sequence is a term. In an arithmetic sequence, each term after the first term (a1) is found by adding a constant, called the common difference (d) to the previous term. + 6 Finding the nth Term ( an ) in an Arithmetic Sequence term 1 2 3 4 5 Common Difference (d) symbols numbers a1 3 + 0(6) a1 + 0(d) a2 9 3 + 1(6) a1 + 1(d) a3 15 3 + 2(6) a1 + 2(d) a4 21 3 + 3(6) a1 + 3(d) a5 27 3 + 4(6) a1 + 4(d) Arithmetic Sequence ...n ... an ...3 + (n – 1)(6) a1 + (n – 1)(d) think
Finding the nth Term ( an ) in an Arithmetic Sequence + 6 Finding the nth Term ( an ) in an Arithmetic Sequence term 1 2 3 4 5 Common Difference (d) symbols numbers a1 3 + 0(6) a1 + 0(d) a2 9 3 + 1(6) a1 + 1(d) a3 15 3 + 2(6) a1 + 2(d) a4 21 3 + 3(6) a1 + 3(d) a5 27 3 + 4(6) a1 + 4(d) Arithmetic Sequence ...n ... an ...3 + (n – 1)(6) a1 + (n – 1)(d) think Find the 11th term in 3, 9, 15, 21, 27, ... an = a1 + (n – 1)(d) a11 = 3 + (11 – 1)(6) a11 = 3 + (10)(6) = ?
Finding the nth Term ( an ) in an Arithmetic Sequence + 6 Finding the nth Term ( an ) in an Arithmetic Sequence term 1 2 3 4 5 Common Difference (d) symbols numbers a1 3 + 0(6) a1 + 0(d) a2 9 3 + 1(6) a1 + 1(d) a3 15 3 + 2(6) a1 + 2(d) a4 21 3 + 3(6) a1 + 3(d) a5 27 3 + 4(6) a1 + 4(d) Arithmetic Sequence ...n ... an ...3 + (n – 1)(6) a1 + (n – 1)(d) think Find the 11th term in 3, 9, 15, 21, 27, ... an = a1 + (n – 1)(d) a11 = 3 + (11 – 1)(6) a11 = 3 + (10)(6) = 63
Finding the nth Term ( an ) in a Geometric Sequence Another type of numerical pattern is a geometric sequence. A geometric sequence is an ordered set of real numbers. Each number in a sequence is a term. In a geometric sequence, each term after the first term ( a1 ) is found by multiplying the previous term by a constant ( r ), called the common ratio. × 2 Finding the nth Term ( an ) in a Geometric Sequence term 1 2 3 4 5 Common Ratio (r) symbols numbers a1 5 × (2)0 a1 × r0 a2 10 5 × (2)1 a1 × r1 a3 20 5 × (2)2 a1 × r2 a4 40 5 × (2)3 a1 × r3 a5 80 5 × (2)4 a1 × r4 Geometric Sequence ...n ... an ... 5 × (2)(n–1) a1 × r(n–1) think
Arithmetic Sequence + 6 Finding the nth Term ( an ) in an Arithmetic Sequence term 1 2 3 4 5 Common Difference (d) symbols numbers a1 3 + 0(6) a1 + 0(d) a2 9 3 + 1(6) a1 + 1(d) a3 15 3 + 2(6) a1 + 2(d) a4 21 3 + 3(6) a1 + 3(d) a5 27 3 + 4(6) a1 + 4(d) Arithmetic Sequence ...n ... an ...3 + (n – 1)(6) a1 + (n – 1)(d) think Find the 11th term in 5, 10, 20, 40, 80, ... an = a1 × r(n – 1) a11 = 5 × (2)10 a11 = 5 (1,024) = ? × 2 Finding the nth Term ( an ) in a Geometric Sequence Common Ratio (r) 5 × (2)0 a1 × r0 10 5 × (2)1 a1 × r1 20 5 × (2)2 a1 × r2 40 5 × (2)3 a1 × r3 80 5 × (2)4 a1 × r4 Geometric ... 5 × (2)(n–1) a1 × r(n–1)
Arithmetic Sequence + 6 Finding the nth Term ( an ) in an Arithmetic Sequence term 1 2 3 4 5 Common Difference (d) symbols numbers a1 3 + 0(6) a1 + 0(d) a2 9 3 + 1(6) a1 + 1(d) a3 15 3 + 2(6) a1 + 2(d) a4 21 3 + 3(6) a1 + 3(d) a5 27 3 + 4(6) a1 + 4(d) Arithmetic Sequence ...n ... an ...3 + (n – 1)(6) a1 + (n – 1)(d) think Find the 11th term in 5, 10, 20, 40, 80, ... an = a1 × r(n – 1) a11 = 5 × (2)10 a11 = 5 (1,024) = 5,120 × 2 Finding the nth Term ( an ) in a Geometric Sequence Common Ratio (r) 5 × (2)0 a1 × r0 10 5 × (2)1 a1 × r1 20 5 × (2)2 a1 × r2 40 5 × (2)3 a1 × r3 80 5 × (2)4 a1 × r4 Geometric ... 5 × (2)(n–1) a1 × r(n–1)
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