Sequence and Series - understanding the ordering of numbers

A set of numbers could be arranged such that they exhibit a definite rule that determines their order. When this happens, those set of numbers are referred to as a Sequence. Such a rule could be addition, subtraction or even the division of a definite whole number. The set of numbers below are a great example of sequence:"

• 10,20,30,40,50....
• 4,16,64,.....
• 3,9,27
• 6,10,14,18,22....

It is interesting to note that in each sequence of numbers above, there is a definite rule which determines the position or value of the succeeding number in each sequence. For example, in the first sequence, the definite rule is that an addition of 10 to the previous number, gives the next number. The same applies to the other set of numbers listed in the above example.

While it might be easy to look at a sequence and determine the next number, it is not always the case. Hence, it is always important to adopt or use a formula already provided to find the next term (nth term) of sequence.
The nth term of a sequence could be denoted thus:

U₁, U₂, U₃, ......nth term (where nth term is the number of unknown figures which make up a sequence) The addition of all the terms of a sequence is known as series. For example, below is the sequence and series of a set of numbers:

Sequence = 15,30,45,60,75....
Series = 15+30+45+60+75

Solve Sequence examples

If U_n = n(1 - 5n) denotes the nth term of a sequence, determine the following:

(i) The first 5 terms of the sequence
(ii) the series of the first 7 terms

Solution

Note that U_n = n(1 - 5n)
For the first term represented by U₁, we have
U₁ = 1(1 - 5x1)
U₁= 1(1 - 5)
U₁= -4

For the 2nd term represented by U₂, we have
U₂ = 2(1 - 5x2)
U₂ = 2(1 - 10)
U₂ = 2(-9)
U₂ = -18

For the 3rd term represented by U₃, we have
U₃ = 3(1 - 5x3)
U₃ = 3(1 - 15)
U₃ = 3(-14)
U₃ = -42

For the 4th term represented by U₄, we have
U₄ = 4(1 - 5x4)
U₄ = 4(1 - 20)
U₄ = 4(-19)
U₄ = -76

For the 5th term represented by U₅, we have
U₅ = 5(1 - 5x5)
U₅ = 5(1 - 25)
U₅ = 5(-24)
U₅ = -120

Therefor, the first five terms of the sequence is -4, -18, -42, -76 -120

(ii) the series of the first 7 terms

Since we have found the first 5 terms, we can just find the 6th and 7th term and then sum them up

For the 6th term represented by U₆, we have
U₆ = 6(1 - 5x6)
U₆ = 6(1 - 30)
U₆ = 6(-29)
U₆ = -174

For the 7th term represented by U₇, we have
U₇ = 7(1 - 5x7)
U₇ = 7(1 - 35)
U₇ = 7(-34)
U₇ = -238

The first seven sequence will is: -4, -18, -42, -76, -120, -174, -238

The series = -4 + (-18) + (-42) + (-76) + (-120) + (-174) + (-238)
Note that when a negative sign multiplies a - sign, the result is negative. Therefore, we have

The series of the first 7 terms = -4 - 18 - 42 - 76 - 120 - 174 - 238
= -672

Conclusion

The nth term of any sequence is very important in determining the number of terms of the sequence. This nth term is usually given as a formula and used to find any number of terms of a sequence. The series is a simple summation of the terms of a sequence. Following the above tips, its easy to solve for any sequence and series. We will see more complex examples in the lesson