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Arithmetic Sequences Name_____________________________ Arithmetic Sequences – we add/subtract to get the next value. 1.) Given the arithmetic sequence, write down what the common difference (d) is. a.) 1, 4, 7, 10, 13, 16, ... d= b.) 8, 18, 28, 38, 48, ... d= c.) 10, 5, 0, -5, -10, ... d= 1 1 d.) 1, , 0, , ... 2 2 d= 5.) Find the 15th number in this sequence: 1, 4, 7, 10, 13, 16, ... 6.) Find the 11th number in this sequence: 8, 18, 28, 38, 48, ... 7.) Find the 6th number in this sequence: 10, 5, 0, -5, -10, ... a7 that 7 is called a subscript a7 is pronounced “a sub 7” 8.) Find the 52nd number in this sequence: 3, 7, 11, 15, 19, 23, … 2.) Fill in the blanks for the sequence 3, 6, 9, 12, 15, 18, 21, 24, 27… a1= ____, a4 = ____, a9 = ____,… a100=______ (bonus) 3.) Fill in the blanks for the sequence Sequences are functions that have the natural numbers as their domain. 9a.) What are the natural numbers? -5, -1, 3, 7, 11, 15, 19, … when n = 1, an = ______________ b.) What would this sequence look like as a table? 100, 200, 300, 400, 500, … when n = 2, an = ______________ when n = 5, an = ______________ 4.) Fill in the blanks for the sequence 10.) Write the first five numbers in the sequence: a n 2n 1 (which is the same as y = 2x + 1) 8, 18, 28, 38, 48, 58, 68, … when n = 5, an = ______________ a1= ____, a2 = ____, a3 = ____, a4= ____, a5 = ____ an+1 = ______________ an+2 = ______________ an-1 = ______________ an-2 = ______________ 11.) Write the first five numbers in the sequence: a1 5 and an an1 3 (Recursive!) a1= ____, a2 = ____, a3 = ____, a4= ____, a5 = ____ Arithmetic Sequences defined explicitly Arithmetic Sequences defined recursively What is the common difference (d)? What is the slope (m)? To define a function recursively: 1.) always give people the 1st term (a1) 2.) Tell people what to do to get the next number using the previous number. What is a 0 ? What is the yintercept? 15.) 12.) 1, 4, 7, 10, 13, 16, ... 4, 7, 10, 13, 16, 19, 22, ... a1 = What is the explicit equation? a n dn a0 an = an-1 + What is the explicit equation? y mx b 16.) 54, 56, 58, 60, 62, 64, … a1 = 13.) 5, 15, 25, 35, 45, ... What is the common difference (d)? What is the slope (m)? What is a 0 ? What is the yintercept? an = an-1 17.) 90, 85, 80, 75, 70, 65,… a1 = What is the explicit equation? a n dn a0 an = What is the explicit equation? y mx b 18.) -7, -5, -3, -1, 1, 3, 5,… 19.) 12, 7, 2, -3, -8, -13,… 14.) 11, 6, 1, -4, -9, ... What is the common difference (d)? What is the slope (m)? What is a 0 ? What is the y-intercept? What is the explicit equation? a n dn a0 What is the explicit equation? y mx b 20.) 16, 18, 20, 22, 24, 26,… Recursive and Explicit 1.) -1, -2, -3, -4, -5, -6, -7, … a64 = a.) Write the explicit equation that represents the sequence: b.) Write the recursive equation that represents the sequence: 2.) 3, 4, 5, 6, 7, 8, 9, ... a64 = a.) Write the explicit equation that represents the sequence: b.) Write the recursive equation that represents the sequence: 3.) 2, 4, 6, 8, 10, 12, ... a64 = a.) Write the explicit equation that represents the sequence: b.) Write the recursive equation that represents the sequence: 4.) 5, 7, 9, 11, 13, 15, 17… a64 = a.) Write the explicit equation that represents the sequence: b.) Write the recursive equation that represents the sequence: Given the recursive definition, write the explicit definition. Given the explicit definition, write the recursive definition. 5.) a1 4 6.) an 2n 1 an an1 3 7.) a1 10 an an1 2 8.) an 3n 9.) Here’s how many push-ups Tony did per day. The pattern continued. Recursive: _____________________________ Recursive: _____________________________ Explicit: ____________________________ Explicit: ____________________________