Problem-Solving Worksheets
Problem Sheet 5 – Sequence and Series Problems
1
Show that, for any natural number n,
a) n(n + 1) is even
b) n3 – n is a multiple of 6
c) n(n + 1)(2n + 1) is a multiple of 6
2
By writing n3 + 11n as n(n2 – 1) + 12n show that
every term of the sequence
n3 + 11n is divisible by 6.
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Show that every term of the sequence
n3 + 5n + 18 is divisible by 6.
3
In a sequence of numbers the nth term is given
by n2 + 2n.
(a) Write the first six numbers in the
sequence;
(b) Explain why the numbers in the sequence
alternate between odd and even.
(c) How many numbers in the sequence are
prime? Explain your reasoning
4
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The natural numbers are entered in a grid in
successive diagonals, as shown. The number 9
has grid position (3, 2). The number 16 has grid
position (1, 6).
What is the grid position of the number 1000?
Problem-Solving Worksheets
5
The first and second terms of an arithmetic
series are 200 and 197.5 respectively. The sum to
n terms of the series is Sn. Find the largest
positive value of Sn.
6
The first three terms of a sequence are
Source: AEA
1+2
3+4+5+6
7 + 8 + 9 + 10 + 11 + 12.
The nth term is the sum of the next 2n natural
numbers that have not already been used to
form a term.
What is the value of the 10th term?
7
The four terms of this sequence are formed such
that each term after the first is the square of the
previous term.
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3x − 2 3x + 2
x − 12,
,
, ax + 6
7
2
What is the value of a?
8
Show that if
1
1
1
,
,
b+c c +a a+b
are three consecutive terms of an arithmetic
series implies then a2, b2 and c2 are also
three consecutive terms of an arithmetic series.
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