Hi there, today I’m going to teach you how to do Arithmetic Sequences in an easy way.
Before we start, we need to know some formula and definition that will help us along the way.
What is sequence?
A succession of numbers, which are arranged in a specific order according to a rule, is called a sequence.
What is term?
Each number in a sequence is called a term of that sequence and in order from the beginning, they are called the 1st term, 2nd term, 3rd term and so on.
The position of each term in a sequence is defined by a natural number n.
For example:
For the 1st term, n=1.
For the 2nd term, n=2.
For the 3rd term, n=3.
Given the sequence 1, 3, 5, 7, 9,11, 13, …
When n=1,the term is 1, which can be expressed as n.
When n=2,the term is 3, which can be expressed as n+1.
When n=3,the term is 5, which can be expressed as n+2.
When n=4,the term is 7, which can be expressed as n+3.
In general, each of the above terms can be expressed as n+(n-1), which becomes 2n-1
1.General term of arithmetic sequence
The arithmetic sequence with 1st term (a) and common difference (d) has the following terms:
A (n)=a+(n-1) d
[EX1]
So for example: we have the sequence: 1, 5, 9, 13, 17, 21…
[SOL] 1st term: a=1
Common difference: d=4, now substitute (a) and (d) into the formula
A (n)=a+(n-1) d =4n-3
And we have 4n-3 as the answer
[EX2]
11th term: -27, common difference: -3
[SOL] Let (a) be the 1st term. And substitute into the formula
A (11) =a+(11-1) x (-3)=-27
A=3
Therefore, the 1st term is 3 and a (n)= -3n+6
[EX3}
1st term: 20 15th term: 90
[SOL] let (d) be the common difference.
A (18)=10+(18-1) d=44
D=2
Therefore, the common difference is 2
[EX4]
7th term: 11 20th term: -41
A (7) =a+6d=11 (1)
A (20)=a+19d=-41 (2)
From (2)-(1)
13d=-52
D=-4
Substituting into (1) a= -48
2.sum of an Arithmetic Sequence
1.S (n) = n (a+l)/2
2.S (n) =n [2a+(n-1) d]/2
[EX1]
A sequence with 1st term 5, last term 16 and number of terms 12.
S (12)= 12(4+16)/2=120
We think 5 is (a) and 16 is the last term (l) and (n) is 12
[EX2]
Expressing l in terms of the 1st term (a) and the common difference (d), l=a+(n-1) d. Substituting this into S (n)= n (a+l)/2 gives the following formula.
5,9,13,17… (Up to the 15th term)
[SOL] 1st term: a=5
Common difference d=4
Number of terms n=15
Substituting S (15)= 15[2x5+(15-1) x4]/2=495
Hope you learnt a lot of information about Arithmetic Sequences.
Additional resources:
Well explained what sequences are, still not an easy subject to understand for me.
everyone feel free to leave a comment ^^