University:
Streetsville Secondary School
Course:
MCR3U (Functions)
Academic year:

2024
Views:

281

Pages:

10
Author:

Jamie G.

CHECK Your Understanding 1. Calculate the sum of the first 10 terms of each arithmetic series. a) 59 + 64 + 69 + ... b) 31 + 23 + 15 + ... c) - 103 - 110 - 117 - ... d) -78 - 56 - 34 - ... 2. Calculate the sum of the first 20 terms of an arithmetic sequence with first term 18 and common difference 11. 3. Bricks are stacked in 20 rows such that each row has a fixed number of bricks more than the row above it. The top row has 5 bricks and the bottom row 62 bricks. How many bricks are in the stack? PRACTISING 4. 1) Determine whether each serics is arithmetic. ii) If the series is arithmetic, calculate the sum of the first 25 terms. a) -5 + 1 + 7 + 13 + ... d) 18 + 22 + 26 + 30 + ... 6) 2 + 10 + 50 + 250 + ... c) 1 +1+2+3+.. e) 31 + 22 + 13 + 4 + ... t) 1-3+5-7 + ... 5. For each series, calculate fiz and S12- Kla) 37 + 41 + 45 + 49 + ... b) -13 - 24 - 35 - 46 - ... с) -18 - 12 - 6 + 0 + ... d) - + 6 5 * 1> + ... 10 e) 3.19 + 4.31 ÷ 5.43 ÷ 6.55 + ... t) p + (2p + 2q) + (3p + 4q) + (4p + 6g) + ... 6. Determine the sum of the first 20 terms of the arithmetic series in which a) the first term is 8 and the common difference is 5 b) 4 = 31 and $20 = 109 c) 4 = 53 and 12 = 37 the 4th term is 18 and the terms increase by 17 e) the 15th term is 107 and the terms decrease by 3 f) the 7th term is 43 and the 13th term is 109 7. Calculate the sums of these arithmetic series. a) 1 + 6 + 11 + ... + 96 d) 5 + 8 + 11 + ... + 2135 b) 24 + 37 + 50 + ... + 349 c) -31 - 38 - 45 c) 85 + 77 ÷ 69 + ... - 99 - ... - 136 f) -63 - 57 - 51 - .. + 63 8. A diagonal in a regular polygon is a line segment joining two nonadjacent A vertices. a) Develop a formula for the number of diagonals for a regular polygon with a sides. b) Show that your formula works for a regular heptagon (a seven-sided polygon). 9. Joe invests $1000 at the start of each year for five ycars and earns 6.3% simple interest on his investments. How much will all his investments be worth at the start of the fifth year? 10. During a skydiving lesson, Chandra jumps out of a plane and falls 4.9 m during the first second. For each second afterward, she continues to fall 9.8 m more than the previous second. After 15 s, she opens her parachute. How far did Chandra fall before she opened her parachute? 11. Jamal got a job working on an assembly line in a toy factory. On the 20th day of work, he assembled 137 toys. He noticed that since he started, every day he assembled 3 more toys than the day before. How many toys did Jamal assemble altogether during his first 20 days? 12. In the video game "Geometric Constructors," a number of shapes have to be arranged into a predefined form. In level 1, you are given 3 min 20 s to complete the task. At each level afterward, a fixed number of seconds are removed from the time until, at level 20, 1 min 45 s are given. What would be the total amount of time given if you were to complete the first 20 levels? 13. Sara is training to run a marathon. The first week she runs 5 km each day. The next week, she runs 7 km each day. During each successive week, each day she runs 2 km farther than she ran the days of the previous week. If she runs for five days each week, what total distance will Sara run in a 10 weck training session? 14. Joan is helping a friend understand the formulas for an arithmetic series. She uses linking cubes to represent the sum of the series 2 + 5 + 8 + 11 + 14 two ways. These representations are shown at the right. Explain how the linking-cube representations can be used to explain the formulas for an arithmetic series. Extending 15. The 10th term of an arithmetic series is 34, and the sum of the first 20 terms is 710. Determine the 25th term. 16. The arithmetic series 1 + 4 + 7 + ... + t, has a sum of 1001. How many terms does the series have? 7.5 Arithmetic Series, pp. 452-453 Question 1: (a): This is an arithmetic series with (n=10, a=59, d=5). (S_{10}=\frac{n[2a+(n-1)d]}{2}) (S_{10}=\frac{10[2(59)+5(10-1)]}{2}=815) (b): This is an arithmetic series with (n=10, a=31, d=-8). (S_{10}=\frac{n[2a+(n-1)d]}{2}) (S_{10}=\frac{10[2(31)+(-8)(10-1)]}{2}=-50) (c): This is an arithmetic series with (n=10, a=-103, d=-7). (S_{10}=\frac{n[2a+(n-1)d]}{2}) Question 1: (c): (S_{10}=\frac{10[2(-103)+(-7)(10-1)]}{2}=-1345) (d): This is an arithmetic series with (n=10, a=-78, d=22). (S_{10}=\frac{n[2a+(n-1)d]}{2}) (S_{10}=\frac{10[2(-78)+(22)(10-1)]}{2}=210) Question 2: For an arithmetic series with (n=20, a=18, d=11), the sum is (S_{20}=\frac{20[2(18)+(11)(20-1)]}{2}=2450). Question 3: The number of bricks in the nth row from the top is (t_{n}). The total number of bricks in the stack is an arithmetic series with (n=20, t_{1}=5, t_{n}=62). (S_{n}=\frac{n(t_{1}+t_{n})}{2}) (S_{20}=\frac{20(5+62)}{2}=670) Question 4: (a): This series is arithmetic since the difference between any two consecutive terms is constant. (b): This series is not arithmetic: the difference between pairs of consecutive terms is not constant. (c): This series is not arithmetic: the difference between pairs of consecutive terms is not constant.