Arithmetic and geometric sequences and series, sum to infinity, sigma notation, recurrence relations, increasing, decreasing and periodic sequences, modelling.
12 original questions, 77 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.
The 5th term of an arithmetic sequence is 23 and the 12th term is 51
Find the first term and the common difference.
Find the sum of the first 20 terms.
4 marks
Show answer
a) a = 7, d = 4b) 900
2
A geometric series has first term 48 and common ratio 34
Find the exact value of the 6th term.
Find the sum of the first 10 terms, giving your answer to 3 significant figures.
Find the sum to infinity.
5 marks
Show answer
a)72964b) 181c) 192
3
Find the value of
30Σr = 1 (5r − 2)
8Σr = 1 3 × 2r
4 marks
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a) 2265b) 1530
4
A sequence is defined by
u1 = 2, un + 1 = kun + 3
where k is a positive constant. Given that u3 = 23
show that 2k2 + 3k − 20 = 0
find the value of k
find 4Σr = 1ur
6 marks
Show answer
a) shownb) k = 52c) 93.5
5
A sequence is defined by
u1 = 3, un + 1 = 11 − un
Find u2, u3 and u4
Describe the behaviour of the sequence.
Find the value of 100Σr = 1ur
6 marks
Show answer
a) u_2 = -12, u_3 = 23, u_4 = 3b) periodic with order 3c)2152
6
The first three terms of a geometric sequence are
k + 3, k − 3, 7 − k
where k is a constant.
Show that k2 − 5k − 6 = 0
Given that the series converges,
find the value of k, giving a reason for your answer,
find the sum to infinity of the series.
6 marks
Show answer
a) shownb) k = 6, because k = -1 gives r = -2 and |r| must be less than 1c)272
7
Priya saves money each month. She saves £40 in the first month and each month after that she saves £5 more than the month before.
Find how much she saves in the 12th month.
Show that the total she has saved after n months is 52n(n + 15) pounds.
Find the month in which her total savings first exceed £3000
6 marks
Show answer
a) £95b) shownc) the 28th month
8
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
A bakery made a profit of £20 000 in its first year of trading, Year 1. A model predicts that the profit will increase by 4% each year, so that the profits form a geometric sequence.
Find the first year in which the predicted profit is more than £30 000
Find the total predicted profit for the first 15 years, giving your answer to the nearest £1000
Give one reason why the model may not be suitable in the long term.
7 marks
Show answer
a) Year 12b) £400 000c) e.g. costs or demand change, so growth will not stay at exactly 4% every year
9
A geometric series has second term 48 and sum to infinity 300
Show that the common ratio, r, satisfies 25r2 − 25r + 4 = 0
Given that r > 12, find the first term of the series.
Find the smallest value of n for which the sum of the first n terms is greater than 299
8 marks
Show answer
a) shownb) a = 60c) n = 26
10
A geometric series has first term a and common ratio r, where r ≠ 1
Prove that the sum of the first n terms of the series is a(1 − rn)1 − r
Hence, or otherwise, show that ∞Σn = 1 12 cos2n (π6) = 36
7 marks
Show answer
S_n = a(1 - rn)/(1 - r) proved by subtracting rS_n from S_n; the sum is a geometric series with a = 9, r = 3/4, so it equals 36
11
Given that nΣr = 1 (4r + 1) = 1430, find the value of n.
Find the value of k for which kΣr = 1 log2 (8r) = 315
Show that 20Σr = 1 ln (3r e2) = 210 ln 3 + 40
8 marks
Show answer
a) n = 26b) k = 14c) shown
12
A fishery stocks a lake with fish. A model for the number of fish, Pn, in the lake at the start of year n is
P1 = 500, Pn + 1 = 0.8Pn + 150
Find P2 and P3
Show that Pn = 750 − 250(0.8)n − 1 satisfies the model.
Explain why the sequence is increasing, and state the long-term number of fish predicted by the model.
Find the first year in which the model predicts more than 740 fish.
10 marks
Show answer
a) P_2 = 550, P_3 = 590b) shownc) 250(0.8)(n - 1) gets smaller each year, so P_n increases; long term 750 fishd) year 16
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Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.