# Sequences and Series

Number sequences and series occur throughout

the study of mathematics and take on a special

variety of applications. One significant

application is in the mathematics of investments

that you will investigate in the final unit.

Sequences occur in science as well. For

example, the number of seeds in each row of a

daisy has a definite pattern. In economics, the

value of invested money can be computed with sequences. A certain sequence, 0.4, 0.7,

1, 1.6, 2.8, 5.2, 10..., lead to the discovery of what was once one of the largest asteroids

in the solar systems, Ceres - with a diameter of 950 km. In 2006, Ceres was redefined as

a dwarf planet by the International Astronomical Union (along with Pluto). And now, the

largest asteroid known is Vesta - which measures 530 km across in diameter.

1. a) What type - arithmetic, geometric or neither - is the sequence that lead to the

discovery of Ceres, formerly the largest asteroid? [1 mark]

b) How do you know if a sequence is arithmetic? [1 mark]

c) How do you know if a sequence is geometric? [1 mark]

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2. The word arithmetic or geometric has been omitted from each of the following

problems. Make a decision as to which type of sequence most likely occurs. State what

type of sequence is involved. Then solve the problem. Be sure to show all of your work.

a) For a sequence, the first term is 2. If the fifth term is 162, find the first three terms of

the sequence. [7 marks]

b) For the sequence 2, 8, 14, ..., what term is equal to 128? [5 marks]

c) The first term of a sequence is 2 and the twentieth term is 40. Find the first four terms.

[4 marks]

d) The sixth term of a sequence is 7 and the tenth term is 19. Find the first term. [6 marks]

3. A simple fractal tree grows in stages. At each new stage, two new line segments

branch out from each segment at the top of the tree. The first five stages are shown.

How many line segments need to be drawn to create stage 20? [4 marks]

4. A sequence has a first term of -3 and each succeeding term is two more than the preceding term.

a) Find the first four terms. [2 marks] b) Find tn. [2 marks]

c) Give a recursive formula [2 marks] d) Find the fiftieth term. [2 marks]

e) Which formula did you use to answer (d), tn or the recursive formula? Why?` [2 marks]

5. Three numbers a, x, and b form an arithmetic sequence, x = 7, and a2+b2 = 148. Find

a, x, and b. [9 marks]

6. An environmental officer has a starting salary of $18 500. Each year the officer is

guaranteed a minimum raise of $1500.00.

a) What will the minimum salary be in 12 years? [3 marks]

b) How much accumulated wealth will the environmental officer have gained, as a

minimum, in 12 years? [4 marks]

7. A stamp is purchased through an investment club for $275.00. The buyer is

guaranteed an increase in value of 12% each year based on the original value.

What is the value of the stamp after the sixth year? [3 marks]

8. Determine the next three terms of the sequence 1, 6, 7, 6, 5, 6, 11, ... Show any work

and/or reasoning [3 marks]

9. A basketball is dropped from a height of 3 m and bounces on the ground. At the top of

each bounce, the ball reaches 60% of its previous height. Provide a neat and well labeled

diagram to represent the situation. Then calculate the total distance travelled by the ball

when it hits the ground for the fifth time. [4 marks]

10. a) Find the first 5 terms of the sequence given by ( ) 3 2 2 2 f n = n − n + and graph the solution.

Provide a sketch of your graph below. [5 marks]

b) Find i) S5 ii) S100 iii) S300 [3 marks]

BONUS: A geometric series has three terms. The sum of the three terms is 42. The third

term is 3.2 times the sum of the other two. What are the terms? [3 marks]

#### Solution Preview

I go through all questions and your solutions. Here are my comments.

Problem #3

If you check the line segments at each stage, we will get sequence as follows.

1, 3, 7, 15, 31

So the rule is t(n) = 2^n - 1, where n is the n-th stage and t(n) is the number of line segments at stage n.

So at stage 20, the number of line segments is t(20) = 2^20 - 1 = ...