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Review Module 2 Investigations 1-4 Name ____________________________Period____ Write the following formulas: Arithmetic– Recursive Formula: ________________________ Explicit Formula: _____________________ Geometric– Recursive Formula: ________________________ Explicit Formula: _____________________ Choose the word or phrase from the list that best matches each sentence. explicit formula recursive formula sequence term 1. an ordered list of numbers or objects __________________________ 2. a formula that describes the nth term of a sequence using the number n __________________________ 3. each number or object in a sequence ____________________ 4. a formula that describes the nth term of a sequence by referring to previous terms ____________________ Suppose you are stacking boxes in levels that form squares. The numbers of boxes in successive levels form a sequence. The figure at the right shows the top four levels as viewed from above. 5. How many boxes are in each of the first four levels? Level 1: Level 2: Level 3: Level 4: 6. How many boxes of equal size would you need for the next lower level? 7. What is an explicit formula that describes the number of boxes in the nth level? 8. How many boxes would you need to add three levels? + + = 9. What is a recursive formula that describes the total number of boxes in a stack of n levels? 10. Suppose you are stacking a total of 91 boxes. Use a formula to find how many levels you will have. 11. You will need levels to make a stack of 91 boxes. Determine whether each formula is explicit or recursive. Then find the first four terms of each sequence. 1 1 12. an = n 13. a1 = 5, an = 3an −1 − 7 14. an = ( n − 1) 3 2 2 15. Consider the sequence: 18, –6, 2, − , . . . 3 a) Describe the pattern. b) Complete the table. Input: term position 1 2 3 4 5 6 c) Graph the sequence. output: term value d) Does the sequence appear to have a limit? If so, what is it? 16. A rubber ball is thrown downward so that it bounces up to a height of 72 inches. After each subsequent 3 bounce, the ball rises to of its previous height. 4 a) Complete the table. Bounce 1 Height (in.) 72 Exponential form 3 72 4 2 3 4 5 54 0 1 3 72 4 b) Recursive Formula:_____________________ Explicit Formula:________________________ c) What height does the ball reach after the 8th bounce? 17. Consider the sequence: –3, 0, 3, 6, . . . Recursive Formula:________________________ Explicit Formula:________________________ 18. Consider the sequence: 2, –6, 18, –54,. . . Recursive Formula:________________________ Explicit Formula:________________________ th 19. Write the explicit formula for the n term of an arithmetic sequence given that a13 = 58 and a28 = 140.5 20. Write the general formulas to determine each of the following: a) Common Difference: d = b) Common Ratio: r = c) Arithmetic Sequence: Recursive Formula:______________________________ Explicit Formula:________________________________ d) Geometric Sequence: Recursive Formula:______________________________ Explicit Formula:________________________________ 21. During your first week of training for a marathon, you run a total of 10 miles. You increase the distance you run each week by 20%. How many miles do you run during your twentieth week of training? a) Describe the pattern. c) What is the first term of the sequence? b) Is the sequence arithmetic, geometric, or neither. d) What is the common ratio of the sequence? e) Write the explicit formula for the sequence. f) Evaluate your formula to find the number of miles you run during your 20th week of training. 22. Farmer Bill purchased a triangular plot of land to grow corn. He arranges the corn seeds in rows. After planting, Farmer Bill sees that in Row 13, he has 58 stalks of corn and in Row 28, Farm Bill counts 118 stalks of corn. Row 1 a. Write an explicit formula for this arithmetic sequence. b. Write a recursive formula for this arithmetic sequence. c. How many corn plants does Farmer Bill have in row 40? d. If Farmer Bill counts 258 stalks of corn in his last row, how many rows of corn does Farmer Bill have? 23. Johnny is taking Pathways Algebra II and needs to get his ducks in a row. He notices that in Row 7, Johnny has 288 ducks. In row 12, Johnny has 9216 ducks. a. Write an explicit formula for this geometric sequence. b. Write a recursive formula for this geometric sequence. c. How many ducks does Johnny have in row 20?