Lesson 3. f(n) = 1 + 4(n 1) or f(n) = n. f(20) = 77. f(n) = 4-7(n 1) or f(n) = 11-7n. f(8) = -45. Exit Ticket Sample Solutions. a.

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1 Exit Ticket Sample Solutions 1. Write the first three terms in the following sequence. Write the explicit formula. a. for and,, f(n) = 9 5(n 1) or f(n) = 14 5n 2. Write an explicit formula for each sequence. a.,,,, pattern, where starts at b.,,,, pattern f(n) =, where starts at Homework Problem Set Sample Solutions S Write a formula for the n th term of the arithmetic sequence 1, 5, 9, 13,. Then use the formula to find f (20). f(n) = 1 + 4(n 1) or f(n) = n f(20) = Find the f(8) of the arithmetic sequence when f(1) = 4 and whose common difference is 7. f(n) = 4-7(n 1) or f(n) = 11-7n f(8) = -45 Unit 7: Sequences 53

2 3. Daniel gets a job with a starting salary of $70,000 per year with an annual raise of $3,000. What will Daniel s salary be in the 10 th year? Write an explicit formula and then solve. S.28 f(n) = (n 1) or f(n) = n f(10) = $34, Save A Lot Theater ticket prices were originally $1 each. Prices have risen by 50 cents each year since. What is the price of a ticket 7 years later? Write an explicit formula and then solve. f(n) = 1 +.5(n 1) or f(n) =.5 +.5n f(7) = $4 5. The graph shows how the cost of a snowboarding trip depends on the number of boarders. A. Fill in the chart of the data B. Write the explicit rule. f(n) = (n 1) or f(n) = n Unit 7: Sequences 54

3 C. Draw a line connecting the data points. What is the y intercept? What is the slope? Write an equation of the line in slope intercept form. The y-intercept is 50 and the slope is 25. y = 25x + 50 D. What do you notice about the answers in Parts B and C? Explain (use m and d in your response). The two equations are essentially the same. The slope of 25 is the same as the common difference of 25 and the y-intercept is found in the explicit formula when you distribute and combine like terms. S Consider the sequence that follows a plus pattern:,,,,,. A. Write a formula for the sequence using the notation. or starting with B. Does the formula generate the same sequence? Why might some people prefer this formula? Yes,. It is nice that the first term of the sequence is a term in the formula, so one can almost read the formula in plain English: Since there is the plus pattern, the th term is just the first term plus that many more threes. C. Graph the terms of the sequence as ordered pairs, on the coordinate plane. What do you notice about the graph? The points all lie on the same line. Unit 7: Sequences 55

4 S Consider a sequence that follows a minus pattern:,,,,. A. Write a formula for the th term of the sequence. Be sure to specify what value of your formula starts with. starting with B. Using the formula, find the th term of the sequence. C. Graph the terms of the sequence as ordered pairs, on the coordinate plane. S.31 CHALLENGE PROBLEMS 8. Find an explicit form for each of the following arithmetic sequences (assume is some real number and is some real number). a.,,,,..., where b.,,,,..., where Unit 7: Sequences 56

5 c.,,,,..., where d.,,,,..., where 9. Consider the arithmetic sequence,,,... a. Find an explicit form for the sequence in terms of., where b. Find the th term. c. If the th term is, find the value of. S If,,,, forms an arithmetic sequence, find the values of,, and. Unit 7: Sequences 57

6 11.,,,... is an arithmetic sequence for some real number. a. Find the value of. The difference between term 1 and term 2 can be expressed as. The difference between term 2 and term 3 can be expressed as. Since the sequence is known to be arithmetic, the difference between term 1 and term 2 must be equal to the difference between term 2 and term 3. Thus,, and ; therefore, the sequence is,,,. b. Find the th term of the sequence., where 12. Find an explicit form of the arithmetic sequence where the nd term is and the sum of the rd term and th term is.,,, Solving this system:,, so, where OR So,. Unit 7: Sequences 58

7 13. In the right triangle figure below, the lengths of the sides cm, cm, and cm of the right triangle form a finite arithmetic sequence. If the perimeter of the triangle is 18 cm, find the values of,, and. Now, do not forget that it is a right triangle, so the Pythagorean theorem must apply:. Since we know that b6, the perimeter equation becomes ac12 once b is substituted. So, substituting c12a and b into the Pythagorean theorem equation gives us a 3612 a, which gives us the following answer: a, b 6, c. Unit 7: Sequences 59