Sequences and Series

WORKSHEET (SEQUENCES AND SERIES) 1.

81 A geometric sequence u1, u2, u3, ... has u1 = 27 and a sum to infinity of 2 . (a)

Find the common ratio of the geometric sequence. (2)

An arithmetic sequence v1, v2, v3, ... is such that v2 = u2 and v4 = u4. N

(b)

Find the greatest value of N such that

∑v n =1

n

>0

. (5) (Total 7 marks)

2.

An arithmetic sequence has first term a and common difference d, d ≠ 0. The 3rd, 4th and 7th terms of the arithmetic sequence are the first three terms of a geometric sequence.

(a)

Show that a =

−

3 d 2 . (3)

(b)

Show that the 4th term of the geometric sequence is the 16th term of the arithmetic sequence. (5) (Total 8 marks)

3.

In the arithmetic series with nth term un, it is given that u4 = 7 and u9 = 22. Find the minimum value of n so that u1 + u2 + u3 + ... + un > 10 000. (Total 5 marks)

4.

Two players, A and B, alternately throw a fair six–sided dice, with A starting, until one of them obtains a six. Find the probability that A obtains the first six. (Total 7 marks)

IB Mathematics Higher Level Worksheet by Dileep

1

IB Mathematics Higher Level Worksheet by Dileep

2

5.

Find the sum of all three-digit natural numbers that are not exactly divisible by 3. (Total 5 marks)

6.

Consider the arithmetic sequence 8, 26, 44, .... (a)

Find an expression for the nth term. (1)

(b)

Write down the sum of the first n terms using sigma notation. (1)

(c)

Calculate the sum of the first 15 terms. (2) (Total 4 marks)

7.

The mean of the first ten terms of an arithmetic sequence is 6. The mean of the first twenty terms of the arithmetic sequence is 16. Find the value of the 15 th term of the sequence. (Total 6 marks)

8.

The sum, Sn, of the first n terms of a geometric sequence, whose nth term is un, is given by

7n − an Sn = (a)

7n

, where a > 0.

Find an expression for un. (2)

(b)

Find the first term and common ratio of the sequence. (4)

IB Mathematics Higher Level Worksheet by Dileep

3

(c)

Consider the sum to infinity of the sequence. (i)

Determine the values of a such that the sum to infinity exists.

(ii)

Find the sum to infinity when it exists. (2) (Total 8 marks)

9.

(a)

Consider the set of numbers a, 2a, 3a, ..., na where a and n are positive integers.

(i) (ii)

a ( n + 1) 2 Show that the expression for the mean of this set is . Let a = 4. Find the minimum value of n for which the sum of these numbers exceeds its mean by more than 100. (6)

(b)

Consider now the set of numbers x1, ... , xm, y1, ... , y1, ... , yn where xi = 0 for i = 1, ... , m and yi = 1 for i = 1, ... , n.

(i)

(ii)

n Show that the mean M of this set is given by m + n and the standard deviation mn S by m + n . Given that M = S, find the value of the median. (11) (Total 17 marks)

10.

5π Consider the graphs y = e and y = e sin 4x, for 0 ≤ x ≤ 4 . –x

(a)

–x

5π On the same set of axes draw, on graph paper, the graphs, for 0 ≤ x ≤ 4 . π Use a scale of 1 cm to 8 on your x-axis and 5 cm to 1 unit on your y-axis. (3)

(b)

nπ Show that the x-intercepts of the graph y = e–x sin 4x are 4 , n = 0, 1, 2, 3, 4, 5. (3)

IB Mathematics Higher Level Worksheet by Dileep

4

(c)

Find the x-coordinates of the points at which the graph of y = e–x sin 4x meets the graph of y = e–x. Give your answers in terms of π. (3)

(d)

(i)

Show that when the graph of y = e–x sin 4x meets the graph of y = e–x, their gradients are equal.

(ii)

Hence explain why these three meeting points are not local maxima of the graph y = e–x sin 4x. (6)

(e)

(i)

Determine the y-coordinates, y1, y2 and y3, where y1 > y2 > y3, of the local maxima

5π of y = e sin 4x for 0 ≤ x ≤ 4 . You do not need to show that they are maximum values, but the values should be simplified. –x

(ii)

Show that y1, y2 and y3 form a geometric sequence and determine the common ratio r. (7) (Total 22 marks)

11.

(a)

The sum of the first six terms of an arithmetic series is 81. The sum of its first eleven terms is 231. Find the first term and the common difference. (6)

(b)

The sum of the first two terms of a geometric series is 1 and the sum of its first four terms is 5. If all of its terms are positive, find the first term and the common ratio. (5)

IB Mathematics Higher Level Worksheet by Dileep

5

(c)

The rth term of a new series is defined as the product of the rth term of the arithmetic series and the rth term of the geometric series above. Show that the rth term of this new series is (r + 1)2r–1. (3)

(d)

Using mathematical induction, prove that n

∑ (r + 1)2

r −1

= n2 n , n ∈

r =1

+

. (7) (Total 21 marks)

12.

A circular disc is cut into twelve sectors whose areas are in an arithmetic sequence. The angle of the largest sector is twice the angle of the smallest sector. Find the size of the angle of the smallest sector. (Total 5 marks)

13.

The common ratio of the terms in a geometric series is 2 x. (a)

State the set of values of x for which the sum to infinity of the series exists. (2)

(b)

If the first term of the series is 35, find the value of x for which the sum to infinity is 40. (4) (Total 6 marks)

14.

(a)

Find the sum of the infinite geometric sequence 27, −9, 3, −1, ... . (3)

IB Mathematics Higher Level Worksheet by Dileep

6

(b)

Use mathematical induction to prove that for n∈

(

+

,

)

a 1− r n . a + ar + ar2 + ... + arn–1 = 1 − r (7) (Total 10 marks)

15.

An 81 metre rope is cut into n pieces of increasing lengths that form an arithmetic sequence with a common difference of d metres. Given that the lengths of the shortest and longest pieces are 1.5 metres and 7.5 metres respectively, find the values of n and d. (Total 4 marks)

16.

A geometric sequence has a first term of 2 and a common ratio of 1.05. Find the value of the smallest term that is greater than 500. (Total 5 marks)

17.

(a)

Show that p = 2 is a solution to the equation p3 + p2 – 5p – 2 = 0. (2)

(b)

Find the values of a and b such that p3 + p2 – 5p – 2 = (p – 2)(p2 + ap + b). (4)

(c)

Hence find the other two roots to the equation p3 + p2 – 5p – 2 = 0. (3)

(d)

An arithmetic sequence has p as its common difference. Also, a geometric sequence has p as its common ratio. Both sequences have 1 as their first term. (i)

Write down, in terms of p, the first four terms of each sequence.

IB Mathematics Higher Level Worksheet by Dileep

7

(ii)

If the sum of the third and fourth terms of the arithmetic sequence is equal to the sum of the third and fourth terms of the geometric sequence, find the three possible values of p.

(iii)

For which value of p found in (d)(ii) does the sum to infinity of the terms of the geometric sequence exist?

(iv)

For the same value p, find the sum of the first 20 terms of the arithmetic sequence, writing your answer in the form a + b c , where a, b, c ∈ . (13) (Total 22 marks)

18.

1 2iθ 1 3iθ e + e 4 Consider the complex geometric series e + 2 + .... iθ

(a)

Find an expression for z, the common ratio of this series. (2)

(b)

Show that │z│ < 1. (2)

(c)

Write down an expression for the sum to infinity of this series. (2)

(d)

(i)

Express your answer to part (c) in terms of sin θ and cos θ.

(ii)

Hence show that

1 1 4 cos θ − 2 cos θ + 2 cos 2θ + 4 cos 3θ + ... = 5 − 4 cos θ . (10) (Total 16 marks)

IB Mathematics Higher Level Worksheet by Dileep

8

19.

A sum of $ 5000 is invested at a compound interest rate of 6.3 % per annum. (a)

Write down an expression for the value of the investment after n full years. (1)

(b)

What will be the value of the investment at the end of five years? (1)

(c)

The value of the investment will exceed $10 000 after n full years. (i)

Write an inequality to represent this information.

(ii)

Calculate the minimum value of n. (4) (Total 6 marks)

IB Mathematics Higher Level Worksheet by Dileep

9