National 5 Maths
Algebraic Patterns
1)
2)
The first 5 terms in a sequence are given below.
Find an expression for the nth term in the sequence.
(a)
3, 5, 7, 9, 11
1
(b)
1, 4, 7, 10, 13
1
(c)
5, 9, 13, 17, 21
1
(d)
20, 30, 40, 50, 60
1
(e)
1, 4, 9, 16, 25
1
(f)
2, 8, 18, 32, 50
1
(g)
2, 5, 10, 17, 26
1
(h)
1, 8, 27, 64, 125
1
(i)
1, 3, 6, 10, 15
1
The sum of consecutive even numbers can be calculated using the following
number pattern:
2+4+6
2+4+6+8
2 + 4 + 6 + 8 + 10
(a)
(b)
(c)
3)
= 3 × 4 = 12
= 4 × 5 = 20
= 5 × 6 = 30
Calculate 2 + 4 + · · · + 20.
Write down an expression for 2 + 4 + · · · + n.
Hence or otherwise calculate 10 + 12 + · · · + 100.
1
1
2
A number pattern is given below.
(a)
(b)
1st term:
22 – 02
2nd term:
32 – 12
3rd term:
42 – 22
Write down a similar expression for the 4th term.
Hence or otherwise find the nth term in its simplest form.
1
2
4)
A number pattern is shown below.
(a)
(b)
(c)
5)
13
=
13 + 23
=
13 + 23 + 33
=
12 × 22
4
22 × 32
4
32 × 42
4
Write down a similar expression for 13 + 23 + 33 + 43 + 53 + 63 + 73.
Write down a similar expression for 13 + 23 + 33 + . . . + n3.
Write down an expression for 83 + 93 + 103 + ….. + n3.
1
2
2
A sequence of terms, starting with 1, is
1, 5, 9, 13, 17, ………..
Consecutive terms in this sequence are formed by adding 4 to the previous
term.
The total of consecutive terms of this sequence can be found using the
following pattern.
Total of the first 2 terms:
Total of the first 3 terms:
Total of the first 4 terms:
Total of the first 5 terms:
(a)
(b)
6)
1+5
1+5+9
1 + 5 + 9 + 13
1 + 5 + 9 + 13 + 17
=2×3
=3×5
=4×7
=5×9
Express the total of the first 9 terms in the same way.
The first n terms of this sequence are added.
Write down an expression, in n, for the total.
2
3
1, 3, 5, 7, ……..
The first odd number can be expressed as
1 = 12 – 02
The second odd number can be expressed as
3 = 22 – 12
The third odd number can be expressed as
5 = 32 – 22
(a) Express the fourth odd number in this form.
(b) Express the number 19 in this form.
(c) Write down a formula for the nth odd number and simplify this
expression.
(d) Prove that the product of 2 consecutive odd numbers is always odd.
1
1
2
3
7)
Brackets can be multiplied out in the following way.
(y + 1)(y + 2)(y + 3) = y3 + (1 + 2 + 3)y2 + (1×2 + 1×3 + 2×3)y + 1×2×3
(y + 2)(y + 3)(y + 4) = y3 + (2 + 3 + 4)y2 + (2×3 + 2×4 + 3×4)y + 2×3×4
(y + 3)(y + 4)(y + 5) = y3 + (3 + 4 + 5)y2 + (3×4 + 3×5 + 4×5)y + 3×4×5
(a)
(b)
8)
In the same way, multiply out
(y + 4)(y + 5)(y + 6)
2
In the same way, multiply out
(y + a)(y + b)(y + c)
2
A number pattern is shown below.
13 + 1 = (1 + 1)(12 – 1 + 1)
23 + 1 = (2 + 1)(22 – 2 + 1)
33 + 1 = (3 + 1)(32 – 3 + 1)
9)
(a)
(b)
(c)
Write down a similar expression for 73 + 1.
Hence write down an expression for n3 + 1.
Hence find an expression for 8p3 + 1.
(a)
Solve the equation
2n = 32
(b)
1
A sequence of numbers can be grouped and added together as shown.
The sum of 2 numbers:
The sum of 3 numbers:
The sum of 4 numbers:
(c)
1
1
2
(1 + 2) = 4 – 1
(1 + 2 + 4) = 8 – 1
(1 + 2 + 4 + 8) = 16 – 1
Find a similar expression for the sum of the first 5 numbers.
1
Find a formula for the first n numbers of this sequence.
2
10)
The sequence of odd numbers starting with 3 is 3, 5, 7, 9, 11, …..
Consecutive numbers from this sequence can be added using the following
pattern.
(a)
(b)
11)
3+5+7+9 =4×6
3 + 5 + 7 + 9 + 11 = 5 × 7
3 + 5 + 7 + 9 + 11 + 13 = 6 × 8
Express 3 + 5 + …….. + 25 in the same way.
The first n numbers in this sequence are added.
Find a formula for the total.
3
The sums, S2, S3 and S4 of the first 2, 3 and 4 natural numbers are given by:
S2 = 1 + 2
S3 = 1 + 2 + 3
S4 = 1 + 2 + 3 + 4
(a)
(b)
12)
2
= ½ (2 × 3) = 3
= ½ (3 × 4) = 6
= ½ (4 × 5) = 10
Find S10 , the sum of the first 10 natural numbers.
Write down the formula for the sum, Sn , of the first n natural
numbers.
1
1
The nth term, Tn of the sequence 1, 3, 6, 10, . . . is given by the formula:
Tn = ½ n(n + 1)
1st term
2nd term
3rd term
(a)
(b)
(c)
13)
T1 = ½ × 1(1 + 1) = 1
T2 = ½ × 2(2 + 1) = 3
T3 = ½ × 3(3 + 1) = 6
Calculate the 20th term, T20.
Show that Tn+1 = ½ (n2 + 3n + 2).
Show that Tn + Tn+1 is a square number.
1
2
2
Using the sequence
1, 3, 5, 7, 9, ……..
(a)
(b)
(c)
Find S3, the sum of the first 3 numbers.
Find Sn, the sum of the first n numbers.
Hence or otherwise, find the (n + 1)th term of the sequence.
1
2
2