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Arithmetic sequence explicit formula allows us to find any term of an arithmetic sequence. An arithmetic sequence can be defined as a sequence of numbers in which the difference between two consecutive numbers is always the same. For example, 3,6,9,12…… is an arithmetic sequence where the first term is 3 and the common difference is (6-3)=(9-6)=(12-9)=3.
The arithmetic sequence explicit formula can be mathematically written as
an = a + (n - 1) d
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Key Terms : Arithmetic sequence formula, first term of the sequence, derivation of arethmetic sequence, common difference in sequence.
Arithmetic Sequence Formula
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Arithmetic sequence explicit formula allows us to find any term of an arithmetic sequence, a1, a2, a3, a4, a5,....., an using its first term (a1) and the common difference (d). This formula will help us to reach the nth term of the sequence. The arithmetic sequence explicit formula can be mathematically written as
an = a + (n - 1)d
The video below explains this:
Arithmetic Progression Detailed Video Explanation:
Also Read: Arithmetic Progression
Derivation of Arithmetic Sequence Formula
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Arithmetic sequence formula can be derived from the terms present in the arithmetic sequence itself. Let us assume the arithmetic sequence is a1, a2, a3, a4, a5,.....,an. Here, the first term which is generally referred to as 'a' is a1. Thus, a=a1. The common difference is 'd' which is the difference between any two adjacent terms of the sequence. It means a2-a1=a3-a2…..= an – an-1. Here, the nth term is representative of the explicit formula of the arithmetic sequence.
an = a + (n - 1) d
Where,
an = nth term of the arithmetic sequence
a= first term of the arithmetic sequence
d= common difference (Difference between every term and its previous term i.e. d= an – an-1).
Also Read: Pascal’s Triangle
Understanding Explicit Formula
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Let us consider an arithmetic sequence of 5,9,13,17,21……… Here, the first term is 5 while the common difference is 4.
We can get any number of the sequence by taking the first term 5 and adding the common difference 4 to it repeatedly. Here is the table to show the calculations for the first few terms.
| n | Calculation for nth term |
|---|---|
| 1 | 5 =5+ 0(4) = 5 |
| 2 | 5+4 = 5+ 1(4) = 9 |
| 3 | 5+4+4 =5+ 2(4) =13 |
| 4 | 5+4+4+4 =5+ 3(4) =17 |
| 5 | 5+4+4+4+4 =5+ 4(4) =21 |
Here, the table shows that we can get the nth term (where n= any number) by taking the first term 5 and adding the common difference 4 repeatedly for (n-1) times. Thus, it can be mathematically written as 5+ 4(n-1).
Generally, this is the standard explicit formula used for an arithmetic sequence whose first term is a and the common difference is d
an = a + (n - 1) d
Also Read:
Examples of Arithmetic Explicit Sequence Formula
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Example 1: Find the 10th term of the arithmetic sequence -3, -1,1, 3……. Using the arithmetic sequence formula.
Solution: Here, the first term of the sequence is -3.
The common difference is -1-(-3) = 2
The 10th term of the given sequence can be calculated by using the formula an = a + (n - 1) d
A10 = -3 + (10-1)2
A10 = -3 + 18
A10 =15
Answer: The 10th term of the given sequence will be 15.
Example 2: Find the common difference of an arithmetic sequence where the first term of the sequence is 0.5 and the 10th term is 9.
Solution: Here, the first term of the sequence is 0.5
10th term is 9.
Let us assume the common difference is ‘d’.
Now, the 10th term of the given sequence can be calculated by using the formula an = a + (n - 1) d
Or, 9= 0.5 + (10-1) d
Or 9d= 8.5
D= 17/18
Answer: the common difference of the given sequence is 17/18.
Also Read:
Things to Remember
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- Arithmetic sequence explicit formula allows us to find any term of an arithmetic sequence, a1, a2, a3, a4, a5,.....,an using its first term (a1) and the common difference (d).
- The common difference refers to the difference between any two adjacent terms of the sequence. It means a2-a1=a3-a2= an – an-1.
- If a1, a2, a3, a4, a5,.....,an is an arithmetic sequence with first term a1 and the common difference is 'd, the nth term is the representative of the explicit formula of the arithmetic sequence, can be mathematically written as
an = a + (n - 1) d
- The arithmetic explicit sequence can be used to find any term of a particular sequence. We can also deduce the common difference of a sequence if the first term and the nth is known and the first term is the common difference and the nth is known.
Also Read: Hyperbola
Sample Questions
Ques. Find out the first three terms in the following sequence. (2 marks)
an = 2n+6
Ans: By substituting n =1,2,3 we will get,
a1= 2(1) +6=8
a2= 2(2) +6=10
a3= 2(3) +6=12
So, the first three terms of the sequence are 8,10,12.
Ques. Find out the 23rd term of the sequence defined by
an = (n-1)(2-n)(n+3) (2 marks)
Ans: Let us assume that the 23rd term will be a23.
Putting n=23 we will obtain
a23 = (23-1)(2-23)(23+3)= 22(-21)(26)=-12012
The 23rd term will be 12012.
Ques. Ajay and Bijay were asked to find an explicit formula for the sequence 26,10,-6,-22,...where the first term should be g(1)
Ajay said the formula should be g(n)= 26-16(n-1)
Bijay said the formula should be g(n)= 42-16n
Which one of them is right? (3 marks)
Ans: Here, the sequence is 26,10,-6,-22……..
The first term is 26 and the common difference is 10-26=-16
The general formula of arithmetic explicit formula an = a + (n - 1) d
Substituting a=26 and d=-16 we will get,
g(n)= 26-16(n-1)
Thus, Ajay was right.
However, if we solve this equation a bit more, we shall get
g(n)= 26-16(n-1)
= 26- 16n+16
=42-16n
Thus, Bijay was right as well.
Both Ajay and Bijay were right.
Ques. Find out an explicit formula for the following arithmetic sequence
{50, 46, 42, 38….} (2 marks)
Ans: In the given sequence the first term is 50 and the common difference is (46-50)=-4
Thus, the explicit formula will be
an= 50 + (n-1)(-4)
= 50 -4n+4
=54-4n
The explicit formula for the sequence will be 54-4n.
Ques. Find an equation for the general term given Find an equation for the general term of the given arithmetic sequence and use it to calculate its 99th term: 7,10,13,16,19,… (3 marks)
Ans. Here, the sequence is 7,10,13,16,19…….
The first term is 7 and the common difference is d= (10-7)=3
an = a1 + (n-1)d
= 7 + (n-1)3
= 3n+4
So, we can write the general term
an= 3n+4…….(i)
Now using the equation (i) we can find out the 99th term.
A99= 3(99)+4= 297+4=301
Therefore, the 99th term will be 301.
Ques. Find the 100th term of the arithmetic sequence -3, 1,5, 9……. Using the arithmetic sequence formula. (2 marks)
Ans: Here, the first term of the sequence is -3.
The common difference is 1-(-3) = 4
The 100th term of the given sequence can be calculated by using the formula an = a + (n - 1) d
A100 = -3 + (100-1)4
= -3 + 396
=393
The 100th term of the given sequence will be 393.
Ques. The general term of a sequence is given by an = -4n + 15. Is the sequence an A. P.? If so, find its 15th term and the common difference. (2 marks)
Ans: an=−4n+15
ak=−4k+15a
ak+1=−4(k+1)+15a
Now
ak+1−ak=−4(k+1)+15−[−4k+15]=−4
Since the difference between two terms is constant. It is a AP
a15=−4(15)+15=−45a15=−4(15)+15=−45
Ques. The 10th and 18th terms of an A. P. are 41 and 73 respectively. Find out the 26th term of the sequence. (2 marks)
Ans: 41=a+9d…(i)
73=a+17d………….(ii)
solving (i) and (ii) we’ll get
a=5,d=4
Thus,
a26=a+25d=105
Ques. The sum of 4th and 8th terms of a sequence is 24 and the sum of 6th and 10th terms is 34. Find the first term and the common difference of the sequence. (2 marks)
Ans. a+3d+a+7d=24
Or,
2a+10d=24…………(i)
a+5d+a+9d=34
Or,
2a+14d=34……….(ii)
Solving equation (i) and (ii) we’ll get
a= -1/2 and d= 5/2
Thus, the first term of the sequence is -1/2 and the common difference is 5/2.
Ques. Find out the second term and nth term of a sequence whose 6th term is 12 and the 8th term is 22. (2 marks)
Ans: a+5d=12………..(i)
a+7d=22…………(ii)
Solving equation (i) and (ii) we’ll get
d=5, a=-13
a2=−13+5=−8
an=−13+(n−1)5=5n−18
Thus, the nth term will be 5n−18.
Ques. Which term of the arithmetic progression 5, 15, 25,………will be 130 more than its 31st term? (2 marks)
Ans: Let’s assume nth term be 130 more than the 31st term
First term = 5
Common difference = 15 - 5 = 10
an = 130 + a31
5 + (n - 1) X 10 = 130 + 5 + (31 - 1) X 10
10 (n - 1) = 430
n = 44
Therefore, the term will be 44.
Ques.If the 8th term of an A. P. is 31 and the 15th term is 16 more than the 11th term, find the sequence. (2 marks)
Ans: a8 = 31
or,
a +(8-1)d = 31
a + 7d = 31….(i)
a15 =16 + a11
Or,
a + 14d = 16 + a +10d
14d = 16 + 10d
or,
4d = 16
or,
d = 4
Now putting d = 4 in equation (i) we get
a + 7(4) = 31
or,
a + 28 = 31
or,
a = 3
Therefore, the sequence will be 3,7,11,15…………
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