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Sequence & Series (HSC)

1) 2)

Yr12-2U\series.hsc Qn1) 2U99-4a

An infinite geometric series has a first term of 8 and a limiting sum of 12. Calculate the common ratio.

Yr12-2U\series.hsc Qn2) 2U98-4b

The third term of an arithmetic series is 32 and the sixth term is 17. i. Find the common difference. ii. Find the sum of the first ten terms. 3)

Yr12-2U\series.hsc Qn3) 2U98-4c

The first term of a geometric series is 16 and the fourth term is i. ii. 4) Find the common ratio. Find the limiting sum of the series.

1 . 4

Yr12-2U\series.hsc Qn4) 2U95-1f

&3 & as a fraction. Express 0 ? 2

5)

Yr12-2U\series.hsc Qn5) 2U94-4d

The positive multiples of 7 are 7, 14, 21, ... . i. What is the largest multiple of 7 less than 1000? ii. What is the sum of the positive multiples of 7 which are less than 1000? 6)

Yr12-2U\series.hsc Qn6) 2U92-10a

i. ii. 7)

For what values of r does the geometric series a + ar + ar2 + ... have a limiting sum? For these values of r write down the limiting sum. 1 1 that has limiting sum . Find a geometric series with common ratio w 1? w

Yr12-2U\series.hsc Qn7) 2U91-5b

The tenth term of an arithmetic sequence is 29 and the fifteenth term is 44. i. Find the value of the common difference and the value of the first term. ii. Find the sum of the first 75 terms. 8)

Yr12-2U\series.hsc Qn8) 2U90-4a

The sum of the first n terms of a certain arithmetic series is given by S n = i. ii. iii. 9) Calculate S1 and S2. Find the first three terms of this series. Find an expression for the nth term.

n (3 n + 1) . 2

Yr12-2U\series.hsc Qn9) 2U89-5a

A geometric series has second term 6 and the ratio of the seventh term to the sixth term is 3. i. Find the common ratio r. ii. What is the first term a? iii. Calculate the sum of the first 12 terms. 10)

Yr12-2U\series.hsc Qn10) 2U88-4a

The first three terms of an arithmetic series are 12, 17 and 22. i. Find the twenty-fifth term of this series. ii. Find the sum of the first twenty-five terms.

1

11)

Yr12-2U\series.hsc Qn11) 2U87-10i

Find the number which when added to each of 2, 6 and 13 will give a set of three numbers in geometric progression. 12)

Yr12-2U\series.hsc Qn12) 2U86-8i

The first and last terms of an arithmetic series are 10 and 60, respectively, and the sum of the series is 3535. Find: a. the number of terms in the series; b. the common difference. 13)

Yr12-2U\series.hsc Qn13) 2U85-7i

a.

b.

For an arithmetic series of first term a and common difference d, write down the formula for the nth term. Use this formula to prove that the sum of the first n terms of this series is n {2 a + (n ? 1) d } . 2 The sum of the first seven terms of an arithmetic series is five times the seventh term. Also the sum of the sixth and seventh terms is 40. Find the sum of the first ten terms of the series.

2

[Answers]

1 3 2) i) –5 ii) 195 1 1 ii) 21 3) i) 4 3 23 4) 99 5) i) 994 ii) 71 071 a 6) i) ?1 < r < 1, 1? r 1 1 1 1 ii) ? ? 2 ? 3 ? 4 ? ... w w w w

1)

7) i) d = 3, a = 2 ii) 8475 8) i) S1 = 2, S2 = 7 ii) 2, 5, 8 iii) 3n ? 1 9) i) 3 ii) 2 iii) 531 440 10) i) 132 ii) 1800 1 11) 3 3 1 12) a) 101 b) 2 13) a) Proof b) 180

3