C2
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SEQUENCES AND SERIES
Worksheet F
The first and fourth terms of a geometric series are 108 and 32 respectively.
a Find the third term of the series.
(5)
b Find the sum to infinity of the series.
(2)
An arithmetic series has first term 6 and common difference 3.
a Find the 20th term of the series.
(2)
Given that the sum of the first n terms of the series is 270,
b find the value of n.
3
(4)
The nth term of a sequence, un , is given by
un = kn − n.
Given that u2 + u4 = 6 and that k is a positive constant,
a show that k =
4
3,
(5)
b show that u3 = 3u1.
(2)
In a computer game, each player must complete the tasks set at each level within a fixed
amount of time in order to progress to the next level.
The time allowed for level 1 is 2 minutes and the time allowed for each of the other levels
is 10% less than that allowed in the previous level.
5
a Find, in seconds, the time allowed for completing level 4.
(2)
b Find, in minutes and seconds, the maximum total time allowed for completing the first
12 levels of the game.
(4)
Evaluate
20
∑ (5r − 1) .
(3)
r =3
6
7
A geometric progression has first term 3 and common ratio −2.
a Find the fifth term.
(2)
b Find the sum of the first ten terms.
(2)
c Show that the sum of the first eight positive terms is 65 535.
(4)
a Prove that the sum, Sn , of the first n terms of an arithmetic progression with first term a
and common difference d is given by
Sn =
1
2
n[2a + (n − 1)d].
(4)
b An arithmetic progression has first term −1 and common difference 7.
Find the largest value of n for which the sum of the first n terms is less than 4000.
8
(5)
The terms of a sequence are defined by the recurrence relation
ur = 2ur–1, r > 1, u1 = 6.
a Write down the first four terms of the sequence.
b Evaluate
10
∑
ur .
(1)
(3)
r =1
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C2
SEQUENCES AND SERIES
9
a State the formula for the sum of the first n natural numbers.
(1)
b Use this formula to find the sum of the natural numbers from 200 to 400 inclusive.
(3)
c Find the smallest value of N for which the sum of the first N natural numbers is greater
than 500 000.
(4)
10
Worksheet F continued
A sequence of terms is given by
un+1 = k + un2, n ≥ 1,
where k is a non-zero constant. Given that u1 = 1,
a find expressions for u2 and u3 in terms of k.
(3)
Given also that u3 = 1,
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b find the value of k,
(3)
c state the value of u25 and give a reason for your answer.
(2)
A sequence is defined by
un+1 = un − 3, n ≥ 1, u1 = 80.
Find the sum of the first 45 terms of this sequence.
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(3)
The first term of a geometric series is t and the sum to infinity of the series is 3t.
a Find the common ratio of the series.
(3)
Given also that the sum of the first four terms of the series is 130,
b find the value of t.
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(4)
The first three terms of an arithmetic progression are t, (2t − 5) and 8.6 respectively.
a Find the value of the constant t.
(2)
b Find the 16th term.
(4)
c Find the sum of the first 20 terms.
(2)
Ginny opens a savings account and decides to pay £200 into the account at the start of each
month. At the end of each month, interest of 0.5% is paid into the account.
a Find, to the nearest penny, the interest paid into the account at the end of the third month. (4)
b Show that the total interest paid into the account over the first 12 months is £79.45 to
the nearest penny.
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(5)
The common ratio of a geometric series is 1.5 and the third term of the series is 18.
a Find the first term of the series.
(2)
b Find the sum of the first six terms of the series.
(2)
c Find the smallest value of k such that the kth term of the series is greater than 8000.
(4)
The sum, Sn , of the first n terms of a series is given by
Sn = 3n − 1.
a Show that the fourth term of the series is 54.
(3)
n
b Show that the nth term of the series can be expressed in the form k(3 ) where k is an
exact fraction to be found.
(4)
c Prove that the series is geometric.
(3)
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