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Sequence and Series

1.       The second term of an arithmetic sequence is 7. The sum of the first four terms of the arithmetic sequence is 12. Find the first term, a, and the common difference, d, of the sequence.

Working:

 

 

Answer:

…………………………………………..

(Total 4 marks)

 

 

2.       The ratio of the fifth term to the twelfth term of a sequence in an arithmetic progression is . If each term of this sequence is positive, and the product of the first term and the third term is 32, find the sum of the first 100 terms of this sequence.

(Total 7 marks)

 


 

3.       An arithmetic sequence has 5 and 13 as its first two terms respectively.

(a)     Write down, in terms of n, an expression for the nth term, an.

(b)     Find the number of terms of the sequence which are less than 400.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 4 marks)

 

 

4.       The sum of the first n terms of an arithmetic sequence is Sn = 3n2  2n. Find the nth term un.

Working:

 

 

Answer:

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(Total 3 marks)

 


 

5.       The probability distribution of a discrete random variable X is given by

P(X = x) = k , for x = 0, 1, 2, ......

          Find the value of k.

Working:

 

 

Answer:

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(Total 3 marks)

 

 

6.       Find the sum of the positive terms of the arithmetic sequence 85, 78, 71, ....

Working:

 

 

Answer:

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(Total 3 marks)

 


 

7.       The sum of an infinite geometric sequence is , and the sum of the first three terms is 13.
Find the first term.

Working:

 

 

Answer:

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(Total 3 marks)

 

 

8.       Find the sum to infinity of the geometric series

Working:

 

 

Answer:

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(Total 3 marks)

 


 

9.       The nth term, un, of a geometric sequence is given by un = 3(4)n+1, n Î +.

(a)     Find the common ratio r.

(b)     Hence, or otherwise, find Sn, the sum of the first n terms of this sequence.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 3 marks)

 


 

10.     Consider the infinite geometric series

(a)     For what values of x does the series converge?

(b)     Find the sum of the series if x = 1.2.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 3 marks)

 


 

11.     Consider the arithmetic series 2 + 5 + 8 +....

(a)     Find an expression for Sn, the sum of the first n terms.

(b)     Find the value of n for which Sn = 1365.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 

 

12.     A sequence {un} is defined by u0 = 1, u1 = 2, un+1 = 3un – 2un–1 where n Î +.

(a)     Find u2, u3, u4.

(3)

(b)     (i)      Express un in terms of n.

(ii)     Verify that your answer to part (b)(i) satisfies the equation
un+1 = 3un – 2un – 1.

(3)

(Total 6 marks)

 


 

13.     A geometric sequence has all positive terms. The sum of the first two terms is 15 and the sum to infinity is 27. Find the value of

(a)     the common ratio;

(b)     the first term.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 

 

 

+.

(a)     Find u2, u3, u4.

(3)

(b)     (i)      Express un in terms of n.

(ii)     Verify that your answer to part (b)(i) satisfies the equation
un+1 = 3un – 2un – 1.

(3)

(Total 6 marks)

 


 

14.     The first four terms of an arithmetic sequence are 2, ab, 2a +b + 7, and a – 3b, where a and b are constants. Find a and b.

Working:

 

 

Answer:

.........................................................................

(Total 6 marks)

 

 

15.     The three terms a, 1, b are in arithmetic progression. The three terms 1, a, b are in geometric progression. Find the value of a and of b given that a ¹ b.

Working:

 

 

Answer:

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(Total 6 marks)

 


 

16.     The diagram shows a sector AOB of a circle of radius 1 and centre O, where = q.

          The lines (AB1), (A1B2), (A2B3) are perpendicular to OB. A1B1, A2B2 are all arcs of circles with centre O.

          Calculate the sum to infinity of the arc lengths

AB + A1B1 + A2B2 + A3B3 + …

Working:

 

 

Answer:

.........................................................................

(Total 6 marks)

 


 

17.     Find an expression for the sum of the first 35 terms of the series

ln x2 + ln¼

          giving your answer in the form ln, where m, n Î .

(Total 5 marks)

 

 

18.     The sum of the first n terms of a series is given by

Sn = 2n2 n, where n Î +.

(a)     Find the first three terms of the series.

(b)     Find an expression for the nth term of the series, giving your answer in terms of n.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 


 

19.     A sum of $5 000 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.

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

(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.

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(Total 6 marks)

 


 

20.     Consider the complex geometric series eiθ e2iθ +e3iθ + …

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

(2)

(b)     Show that ½z½< 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

cos θ +cos 2θ + cos 3θ +…=

(10)

(Total 16 marks)

 

 

21.     The sum of the first n terms of an arithmetic sequence {un} is given by the formula
Sn = 4n2 – 2n. Three terms of this sequence, u2, um and u32, are consecutive terms in a geometric sequence. Find m.

(Total 6 marks)

 


 

22.     In an arithmetic sequence the second term is 7 and the sum of the first five terms is 50. Find the common difference of this arithmetic sequence.

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(Total 6 marks)

 


 

23.     The sum to infinity of a geometric series is 32. The sum of the first four terms is 30 and all the terms are positive.

Find the difference between the sum to infinity and the sum of the first eight terms.

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(Total 6 marks)

 


 

24.     A sum of $100 is invested.

(a)     If the interest is compounded annually at a rate of 5% per year, find the total value V of the investment after 20 years.

(b)     If the interest is compounded monthly at a rate of % per month, find the minimum number of months for the value of the investment to exceed V.

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(Total 6 marks)

 


 

25.     An infinite geometric series is given by

(a)     Find the values of x for which the series has a finite sum.

(b)     When x = 1.2, find the minimum number of terms needed to give a sum which is greater than 1.328.

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(Total 6 marks)

 


 

26.     Consider the arithmetic series −6 +1 +8 +15 +....

Find the least number of terms so that the sum of the series is greater than 10 000.

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(Total 6 marks)

 


 

27.     The first and fourth terms of a geometric series are 18 and  respectively.

Find

(a)     the sum of the first n terms of the series;

(4)

(b)     the sum to infinity of the series.

(2)

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(Total 6 marks)

 


 

28.     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.

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(Total 5 marks)

 


 

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

(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.

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(4)

(Total 6 marks)

 

 

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

(3)

(b)     Use mathematical induction to prove that for nÎ+,

                   a + ar + ar2 + ... + arn–1 =

(7)

(Total 10 marks)

 

 

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