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Notes PICTURE AND NUMBER PATTERNS Complete the following picture patterns by finding the next term(s). 1. 2. ___ ______ _________ 3. ___ ______ _________ _____ 5. ______ _________ _____ _____________ 50th shape 100th shape _____________ _____________ ___________ _____________ 50th shape 100th shape _____________ _____________ ___________ 4. ___ _____ _______ ___________ 50th shape 100th shape ___________ ___________ ___________ _____________ 50th shape 100th shape ___ ______ _________ _____________ _____________ ___________ 6. Underneath the picture patterns in # 2, 3, 4, and, 5 write a corresponding number pattern. 7. Find the number underneath the 50th and 100th shape for # 2, 3, 4, and 5. 8. Complete the following number patterns by finding the missing terms. a. 5, 7, 9, 11, 13, 15, 17, __________, _________, __________ b. -4, -9, -14, -19, -24, -29, -34, __________, _________, __________ c. 405, 135, 45, 15, __________, _________, __________ d. 5, 6, 8, 11, __________, _________, __________, 33, __________, 50, __________ e. 2, 6, 18, 54, ________, __________, 1458, __________ f. 2, 3, 5, 8, 13, 21, __________, _________, __________, 144, __________ g. 3, 9, 27, 81, __________, __________, 2187, ___________, __________ Underneath each number patterns above write if the sequence is arithmetic, geometric or neither. Arithmetic Sequence = a sequence where you add or subtract the same number to get the next term. Geometric Sequence = a sequence where you multiply or divide by the same number to get the next term. Neither = all the other sequences! ©Dr Barbara Boschmans Page 1 of 4 Notes 9. Use the rule to determine the first eight terms of each sequence. Example: Each term is the term number times 3 plus 2. First Term 1×3+2 5 Second Term 2×3+2 8 Third Term 3×3+2 11 Fourth Term 4×3+2 14 Fifth Term 5×3+2 17 Etc. a) Each term is the term number times 4 plus 1. 1 2 3 4 5 6 7 8 ___ ___ 13 ___ ___ ___ ___ 33 b) Each term is the term number times 6 minus 3. 1 2 3 4 5 6 7 8 ___ ___ ___ ___ 27 ___ ___ ___ c) Each term is the term number times – 4 plus 50. 1 2 3 4 5 6 7 8 46 ___ ___ ___ ___ ___ ___ ___ d) Each term is the term number squared plus 3. 1 2 3 4 5 6 7 8 ___ ___ ___ ___ ___ 39 ___ ___ e) Each term is the term number times the next term number. 1 2 3 4 5 6 7 8 ___ ___ ___ 20 ___ ___ ___ ___ f) Find the differences between the successive terms in #1-3. What do you notice? What do we call this type of sequence? g) If you did not know the rules for the sequences in # 4-5, how could you find the next five terms of each sequence? 10. Let’s use this example: 5, 8, 11, 14, 17, … Organizing this information in a table will be useful: Term Number 1 2 3 4 5 Term 5 8 11 14 17 Difference ___ ___ What is the constant difference? ______ ©Dr Barbara Boschmans 6 7 8 ___ ___ ___ ___ ___ Page 2 of 4 Notes You need to find a general equation for this sequence (also called the “nth” term) and use it to find the 100th term. Let’s help you out: Term Number Constant Difference What Was Done? To Get 1 3 3 _____ = 5 1st Term 2 3 6 _____ = 8 2nd Term 3 3 _____ _____ = 11 3rd Term 4 ___ _____ _____ = 14 4th Term 5 ___ _____ _____ = 17 5th Term n ___ _____ _____ = _______ nth Term 100 ___ _____ _____ = _______ 100th Term General Equation / nth term / rule: ____________ 100th Term: ________ Find the rule/nth term for each of the following sequences and use your rule to find the 25th and 100th term. nth Term 25th Term 100th Term a. 9, 14, 19, 24, 29, ___, ___, ___, …, __________ __________ _________ b. -3, 4, 11, 18, 25, ___, ___, ___, …, __________ __________ _________ c. -1, -4, -7, -10, -13, ___, ___, ___, …, __________ __________ _________ d. 104, 102, 100, 98, 96, ___, ___, ___, …, __________ __________ _________ 11. Find the sum for each of the following arithmetic sequences. a. 9, 14, 19, 24, 29, … , 499, 504 Sum = 9 + 14 + 19 + … + 499 + 504 = ______________________ b. -3, 4, 11, 18, 25, … , 683, 690 Sum = -3 + 4 + 11 + … + 683 + 690 = ______________________ c. -1, -4, -7, -10, -13, … , -295, -298 Sum = -1 + -4 + -7 + … + -295 + -298 = _____________________ d. 104, 102, 100, 98, 96, … , -92, -94 Sum = 104 + 102 + 100 + … + -92 + -94 = _________ ©Dr Barbara Boschmans Page 3 of 4 Notes 12. Let’s use this example: 2, 6, 18, 54, 162, … Organizing this information in a table will be useful: Term Number 1 2 3 4 5 Term 2 6 18 54 162 6 7 8 Factor ___ ___ ___ ___ ___ ___ ___ What is the factor? ______ You need to find a general equation for this sequence (also called the “nth” term) and use it to find the 100th term. Let’s help you out. This time we will start with our numbers in the sequence and “peel” them apart. Sequence Number First Term Factored Out Re-write using the Factor Term Number Term Number Minus One 2 =2 = 2 x 30 1 0 1st Term 6 =2x3 = 2 x 31 2 1 2nd Term 18 =2x9 = 2 x 32 3 2 3rd Term 54 = 2 x 27 = 2 x 33 4 3 4th Term 162 = 2 x 81 = 2 x 34 5 4 5th Term _______ _______________ n n-1 nth Term _______ _______________ 100 99 100th Term = = General Equation / nth term / rule: ____________ 100th Term: ________ Find the rule/nth term for each of the following sequences and use your rule to find the 25th and 100th term. nth Term 25th Term 100th Term a. 400, 200, 100, 50, 25, ___, ___, ___, …, __________ __________ _________ b. -1, -5, -25, -125, -625, ___, ___, ___, …, __________ __________ _________ c. 4, 8, 16, 32, 64, ___, ___, ___, …, __________ __________ _________ d. 64, -32, 16, -8, 4, ___, ___, ___, …, __________ __________ _________ ©Dr Barbara Boschmans Page 4 of 4