2.1 Exercises in the Sigma Notation

Before we generalize the indexed notation, let us give some examples with the sigma form.

A notation i=15x i means that i is replaced by whole numbers starting with 1 until the number 5 is reached. Thus

∑4
    xi = x2 + x3 + x4
i=2

and

∑ 5
    xi = x2 + x3 + x4 + x5
 i=2

Hence, the notation i=1n tells us:

  1. to add the numbers xi,

  2. to start with i = 1, that is, with x1,

  3. to stop with i = n, that is, with xn (where n is some positive integer).

As an example, let us assign the following values: x1 = 10, x2 = 8, x3 = 2, x4 = 15, and x5 = 22. Then we have:

∑5
    xi = x1 + x2 + x3 + x4 + x5 = 10 + 8 + 2 + 15 + 22 = 57
i=1

The name i is a dummy variable, any other name could be used. We could have used j, the expression would have be exactly the same, hence:

∑5       ∑5
    xi =    xj
i=1      j=1

Now let us find i=143x i based on the previous values. Again we start with i = 1 and we replace 3xi with its value:

∑4
    3xi = 3x1 + 3x2 + 3x3 + 3x4 = 3 ⋅ 10 + 3 ⋅ 8 + 3 ⋅ 2 + 3 ⋅ 15 = 105
i=1

Similarly, let use find i=25(x i - 8): This is:

∑ 5
    (xi - 8) = (x2 - 8) + (x3 - 8) + (x4 - 8 ) + (x5 - 8)
 i=2
     = (8 - 8) + (2 - 8) + (15 - 8) + (22 - 8) = 15

One should be careful with the parentheses. The expression i=25(x i - 8) is not the same as i=25x i - 8. The later evaluates to (the 8 is not included in the sum):

∑5
    xi - 8 = x2 + x3 + x4 + x5 - 8 = 39
i=2

We also use sigma notation in the following way:

∑4
    j2 = 12 + 22 + 32 + 42 = 30
j=1

The same principle applies here. j is replaced in the expression j2 by numbers starting with 1 and ending with 4, and then adding up all four terms.

For the sigma notation we have three important transformation rules:

Rule 1: if c is a constant, then:

∑n         ∑n
    cxi = c    xi
i=1        i=1

Rule 2: if c is a constant, then:

∑n
    c = nc

i=1

Rule 3: Adding the term can be distributed:

∑n            ∑n       ∑n
   (xi + yi) =     xi +    yi
i=1            i=1      i=1

Proof:

i=1ncx i = cx1 + cx2 + + cxn-1 + cxn
= c (x1 + x2 + + xn-1 + xn) = c i=1nx i
i=1nc = c◟ +-c +◝.◜..+-c◞n-times = n × c = nc
i=1n(x i + yi) = (x1 + y1) + + (xn + yn)
= (x1 + + xn) + (y1 + + yn) = i=1nx i + i=1ny i

End of proof