Pure: Sequences, Series and Binomial Depth: Fluency and Exam Drill - Worksheets, Questions and Revision

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A-Level · Pure Mathematics

P12D Pure: Sequences, Series and Binomial Depth: Fluency and Exam Drill

EDEXCEL 9MA0 · Calculator allowed · about 50 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

An arithmetic sequence has first term 5 and common difference 3. Find the 6th term.

(Total for Question 1 is 1 mark)

Find the sum of the first 10 terms of an arithmetic series with first term 4 and common difference 6.

(Total for Question 2 is 1 mark)

A geometric sequence has first term 3 and common ratio 2. Find the 5th term.

(Total for Question 3 is 1 mark)

Find the sum of the first 4 terms of a geometric series with first term 2 and common ratio 3.

(Total for Question 4 is 1 mark)

Find the coefficient of x² in the binomial expansion of (1+x)⁵.

(Total for Question 5 is 1 mark)

A geometric series has first term 8 and common ratio 0.5. State the sum to infinity.

(Total for Question 6 is 1 mark)

Write down the first three terms, in ascending powers of x, of the binomial expansion of (1+x)⁴.

(Total for Question 7 is 1 mark)

An arithmetic sequence has first term 10 and 4th term 22. State the common difference.

(Total for Question 8 is 1 mark)

State, with a reason, whether the geometric series with first term 5 and common ratio 1.2 has a sum to infinity.

(Total for Question 9 is 1 mark)

State the coefficient of x in the binomial expansion of (1+2x)³.

(Total for Question 10 is 1 mark)

An arithmetic sequence has 3rd term 11 and 8th term 31. Find the common difference and the first term.

(Total for Question 11 is 2 marks)

Find the sum of the first 20 terms of the arithmetic series with first term 7 and common difference -2.

(Total for Question 12 is 2 marks)

A geometric sequence has first term 4 and 4th term 32. Find the common ratio, given that it is positive.

(Total for Question 13 is 2 marks)

Find the coefficient of x³ in the binomial expansion of (1-x)⁶.

(Total for Question 14 is 2 marks)

An arithmetic series has first term 6 and common difference 5.

(a)Find the 20th term.(2)

(b)Find the sum of the first 20 terms.(2)

(Total for Question 15 is 4 marks)

A geometric series has first term 10 and common ratio -0.5. Find the sum to infinity, giving your answer as an exact fraction.

(Total for Question 16 is 3 marks)

A sequence is defined by u1 = 20 and u_(n+1) = 0.5 u_n + 3, for n ≥ 1.

(a)Find the values of u2 and u3.(2)

(b)Show that the sequence with terms u_n - 6 is a geometric sequence, and state its common ratio.(2)

(Total for Question 17 is 4 marks)

g(x) = (1+3x)⁴ (1-x)^-1, for |x| < 1/3.

(a)Find the binomial expansion of (1+3x)⁴ in ascending powers of x, up to and including the term in x².(2)

(b)Given that (1-x)^-1 = 1 + x + x² + ... for |x| < 1, hence show that g(x) = 1 + 13x + 67x² + ... for small values of x.(2)

(Total for Question 18 is 4 marks)

The sum of the first n terms of an arithmetic series is given by S_n = 2n² + 7n. Find the first term and the common difference of the series.

(Total for Question 19 is 3 marks)

f(x) = 1/√1+4x = (1+4x)^-1/2, for |x| < 1/4.

(a)Find the binomial expansion of f(x) in ascending powers of x, up to and including the term in x², giving each coefficient as an exact number.(3)

(b)By substituting x = 0.01 into the expansion from part (a), find an estimate for 1/√1.04, giving your answer to 4 decimal places.(1)

(Total for Question 20 is 4 marks)

Mark scheme · P12D Pure: Sequences, Series and Binomial Depth: Fluency and Exam Drill

Question 1

B1 20 cao
Answer: 20

Question 2

B1 310 cao
Answer: 310

Question 3

B1 48 cao
Answer: 48

Question 4

B1 80 cao
Answer: 80

Question 5

B1 10 cao
Answer: 10

Question 6

B1 16 cao
Answer: 16

Question 7

B1 1 + 4x + 6x², all three terms correct
Answer: 1 + 4x + 6x²

Question 8

B1 d = 4 cao
Answer: 4

Question 9

B1 no, since |r| = 1.2 > 1
Answer: No, since |r| > 1

Question 10

B1 6 cao
Answer: 6

Question 11

M1 forms 5d = 31 - 11 = 20 (or equivalent simultaneous equations)
A1 d = 4 and a = 3
Answer: d = 4, a = 3

Question 12

M1 substitutes into S_n = (n/2)(2a + (n-1)d)
A1 -240 cao
Answer: -240

Question 13

M1 forms r³ = 32/4 = 8
A1 r = 2 cao
Answer: r = 2

Question 14

M1 identifies the term as C(6,3)x³(-1)³
A1 -20 cao
Answer: -20

Question 15

(a) M1 substitutes into a + 19d
(a) A1 101 cao
(a) Answer: 101
(b) M1 substitutes into S_n = (n/2)(2a+(n-1)d)
(b) A1 1070 cao
(b) Answer: 1070

Question 16

M1 substitutes a = 10, r = -0.5 into a/(1-r)
A1 = 10/1.5
A1 = 20/3 cao (exact)
Answer: 20/3

Question 17

(a) M1 substitutes u1 = 20 into the recurrence
(a) A1 u2 = 13, u3 = 9.5
(a) Answer: u2 = 13, u3 = 9.5
(b) M1 u_(n+1) - 6 = 0.5u_n + 3 - 6 = 0.5(u_n - 6)
(b) A1 cso: common ratio = 0.5, confirming the sequence is geometric
(b) Answer: Common ratio 0.5

Question 18

(a) M1 uses the binomial expansion with terms C(4,1)(3x) and C(4,2)(3x)²
(a) A1 1 + 12x + 54x²
(a) Answer: 1 + 12x + 54x²
(b) M1 multiplies (1 + 12x + 54x²) by (1 + x + x²), collecting terms up to x²
(b) A1 cso: 1 + 13x + 67x² as required
(b) Answer: g(x) = 1 + 13x + 67x² + ...

Question 19

M1 finds S1 = 9 (the first term) and S2 = 22
A1 second term = S2 - S1 = 13, so d = 13 - 9 = 4
A1 a = 9 cao
Answer: a = 9, d = 4

Question 20

(a) M1 uses the binomial series with n = -1/2 and u = 4x
(a) A1 coefficient of x = -2
(a) A1 coefficient of x² = 6, giving 1 - 2x + 6x²
(a) Answer: f(x) = 1 - 2x + 6x² + ...
(b) B1 0.9806 (ft from (a))
(b) Answer: 0.9806

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Pure: Sequences, Series and Binomial Depth: Fluency and Exam Drill - Worksheets, Questions and Revision

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