MAS004 Sheet 11: Sums and products
Friday, 20th March
Notes in brief
Quite a lot of mathematics, beginning with what you did in primary school,
involves adding an organised list of things up, or multiplying them together.
It’s very useful to have notation for this. So we invented some!
Suppose we wish to write the sum
12 + 22 + 32 + 42 + · · · + 992 + 1002 .
We use notation based on the kind of thing we’re adding up, namely n2 for various values of n, and the range of values we use to form that kind of thing, namely
P100
n running from 1 to 100. In our notation, this is represented as n=1 n2 . We
read this example out loud as
The sum, as n ranges from 1 to 100, of n2 .
The symbol is a Greek capital letter sigma. The little numbers represent the
range: they’re written above and below the Σ sign if we have space, and to the
right (as we’ve done there) if not. The things we’re adding are written after
that.
Similarly, we write
4
X
3n = 31 + 32 + 33 + 34 = 3 + 9 + 27 + 81 = 120.
n=1
Note that (as for integrals), the variable we use is our choice. We have
10
X
n=1
n2 =
10
X
m=1
m2 =
10
X
k2 =
k=1
10
X
α=1
α2 =
10
X
Q2 .
Q=1
The only thing that’s important is that we must not use a variable that already
has some other meaning
Pn in the problem. So if we’re trying to capture 1+· · ·+n,
we must not write n=1 n, because it makes no sense to ask for n to vary
between 1 and n.
Products have very similar notation to sums, except we use a large capital
pi, rather than a large capital sigma:
5
Y
(2k + 1) = (2 × 1 + 1)(2 × 2 + 1)(2 × 3 + 1)(2 × 4 + 1)(2 × 5 + 1)
k=1
= 3 × 5 × 7 × 9 × 11 = 10395.
1
The best-known example of a product is one we have a special name, and
special notation for. The factorial of n, written n! is the product of all numbers
from 1 to n:
n
Y
n! =
i
i=1
It’s well worth remembering the first few values of this:
1! = 1,
2! = 2,
3! = 6,
4! = 24,
5! = 120,
6! = 720.
Exercises
Exercise 1.
Write these out in full, and then evaluate each of them:
P4
P7
3
(a)
(e)
m=1 m ;
m=3 m(m − 1);
P8
2
(b)
m=2 m ;
P9
(f)
P5
m=3 2;
2
(c)
m=1 (m + m);
P4
m 2
9
(d)
(g) πm=3
2.
m=1 (−1) m ;
Exercise 2.
P
Write the following using the -notation:
(a) 1 + 2 + 3 + 4 + · · · + 150;
(b) 1 +
1
2
+
1
3
+ · · · + n1 ;
(c) 14 + 24 + 34 · · · + n4 + (n + 1)4 ;
(d) (10 × 14) + (11 × 15) + (12 × 16) + (13 × 17) + (14 × 18) + · · · + (100 × 104);
(e) 13 + 33 + 53 + · · · + 993 .
Exercise 3.
Evaluate the following without using a calculator. (It helps to do a lot of cancellation in most of them).
(a) 7!;
(b)
8!
2!6! ;
(c)
8!
3!5! ;
(d)
8!
(4!)2 .
Exercise 4.
Pn
For which n do we have k=1 k 3 = 3025?
Exercise 5.
Produce a general formula for
Qn
k=1 (1
k
+ x2 ).
2
Solutions
Solution 1.
(a) 13 + 23 + 33 + 43 = 1 + 8 + 27 + 64 = 100;
(b) 22 + 32 + 42 + 52 + 62 + 72 + 82 = 203;
(c) (12 + 1) + (22 + 2) + (32 + 3) + (42 + 4) + (52 + 5) = 70;
(d) −12 + 22 − 32 + 42 = 10;
(e) 3 × 2 + 4 × 3 + 5 × 4 + 6 × 5 + 7 × 6 = 110.
Solution 2.
For all of these, the answers given are not unique, and other examples may be
entirely appropriate.
P150
(a)
n=1 n;
Pn 1
(b)
i=1 i ;
Pn+1 4
(c)
i=1 i ;
P100
(d)
k=10 k(k + 4);
P50
3
(e)
k=1 (2k − 1) .
Solution 3.
(a) 5040;
(b) 28;
(c) 56;
(d) 70.
Solution 4.
It’s true for n = 10.
Solution 5.
It is equal to 1 + x + x2 + · · · + x2
k+1
−1
k+1
=
3
1−x2
1−x
.