Arithmetic sequence explicit formula is useful to find any terms of the given arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the differences between anytwo consecutive numbers are thesame. For an arithmetic sequence 2, 5, 8, 11, .... the first term is a = 2, and the common difference is d = 5 - 2 = 3. The arithmetic sequence explicit formula for this series is an= a + (n - 1)d, or an= 2 + (n - 1)3 or an = 3n - 1.
Here the arithmetic sequence explicit formula (an= 3n - 1) is useful to find any terms of the series and can be calculated without knowing the previous term. Let us learn the arithmetic sequence explicit formula, and its derivation with the help of examples, FAQs.
What Is ArithmeticSequenceExplicitFormula?
The arithmetic sequence explicit formula is used to find any term (nthterm) of the arithmetic sequence, a1, a2, a3, ..., an,....using its first term (a) and the common difference (d). This formula givesthe nthterm formula of an arithmetic sequence. The arithmetic sequence explicit formula is an= a + (n - 1)d.
Derivation of ArithmeticSequenceExplicitFormula
The arithmetic sequence explicit formula is derived from the terms of the arithmetic sequence. It helps to easily find any term of the arithmetic sequence. The arithmetic sequence isa1, a2, a3, ..., an. Here the first term is referred as 'a' and we have a = a1 and the common difference is denoted as'd'. The formula for the common difference is d = a2- a1= a3- a2= an- an - 1. The nthterm of the arithmetic sequence represents the explicit formula of the arithmetic sequence.
Explicit Formula: an= a + (n - 1) d
where,
- \(a_n\) = nthterm of the arithmetic sequence
- a = the first term of the arithmetic sequence
- d = the common difference (the difference between every term and its previous term. i.e., d \(=a_n-a_{n-1}\)
Example of Arithmetic Sequence Explicit Formula
Anexample of an arithmetic sequence is-3, -6, -9, -12, ... , for whichthe difference between anyterm and its previous term is -3. This difference is usually termedas the common difference and is denoted by d. The first term of an arithmetic sequence is represented by a1or a = -3. Applying the formula for the nthterm, the arithmetic series explicit formula is an= a + (n - 1)d = -3 + (n - 1)(-3) = -3n + 3 - 3 = -3n, or an= -3n.
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Examples UsingArithmeticSequenceExplicitFormula
Example 1: Find the 15thterm of the arithmetic sequence -3, -1, 1, 3, ..... using the arithmetic sequence explicit formula.
Solution:
The first term of the given sequence is, a = -3
The common difference is, d = -1 - (-3) (or) 1- (-1) (or) 3 - 1 = 2.
The 15thterm of the given sequence is calculated using,
\(a_n\) = a + (n - 1) d
\(a_{15}\) = -3 + (15 - 1) 2 = -3 + 14(2) = -3 + 28 = 25.
Answer: The 15thterm of the given sequence = 25.
Example 2:Find the common difference of the arithmetic sequence whose first term is 1/2 and whose 10thterm is 9.
Solution:
The first term is, a = 1/2.
Its 10thterm is a\(_{10}\) = 9.
Using thearithmetic sequence explicit formula, we have the following formula.
\(a_n\) = a + (n - 1) d
Substituting n = 10, we get
\(a_{10}=a+(n-1)d\)
9 = (1/2) + (10 - 1) d
9 = (1/2) + 9d
Subtracting 1/2 from both sides,
17/2 = 9d
Dividing both sides by 9,
d = 17/18
Answer:The common difference is 17/18.
Example 3:Find the general term (or) nthterm of the arithmetic sequence -1/2, 2, 9/2, .....
Solution:
The first term of the given sequence is, a = -1/2.
The common difference is d = 2 - (-1/2) (or) 9/2 - 2 = 5/2.
We can find the general term (or) nthterm of an arithmetic sequence using thearithmetic sequence explicit formula.
\(a_n\) = a + (n - 1) d
\(a_n\) = -1/2 + (n - 1) (5/2)
\(a_n\) = - 1/2+ 5/2n -5/2
\(a_n\) = (5/2) n - 3
Answer: The nthterm of the given sequence = (5/2) n - 3.
FAQs onArithmeticSequenceExplicitFormula
What IsArithmeticSequenceExplicitFormula?
The arithmetic sequence explicit formula is a formula that is used to find the nthterm of an arithmetic sequence without computing any other terms before the nthterm. Using this formula, the nthterm of an arithmetic sequence whose first term is 'a' and common difference is 'd' is, a\(_n\) = a + (n - 1) d.
How To DeriveArithmeticSequenceExplicitFormula?
We know that the differences between every two consecutive terms of an arithmetic sequence are all same. Thus, an arithmetic sequence is of the form a, a + d, a + 2d, .... If we observe here, the 1stterm is a = a + (1 - 1) d, the 2ndterm is a + d = a + (2 - 1) d, and the 3rdterm is a + 2d = a + (3 - 1) d. In the same way, the nthterm is,a\(_n\) = a + (n - 1) d.
What Are the Applications ofArithmeticSequenceExplicitFormula?
Using the arithmetic sequence explicit formula, we can find any term of an arithmetic sequence just with the help of the first term and the common difference. For example, to find the 50thterm of an arithmetic sequence -7, -5, -3, ..., we just use the first term, a = -7 and the common difference, d = 2 and substitute in the formulaa\(_n\) = a + (n - 1) d, then we geta\(_{50}\) = (-7)+ (50- 1) 2 =91.
What Is the 100thterm of the Arithmetic Sequence 1/4, 1/2, 3/4, ... UsingArithmeticSequenceExplicitFormula?
In the given sequence, a = 1/4 and the common difference is, d = 1/2 - 1/4 = 3/4 - 1/2 = ... = 1/4. Substituting n = 100 in the arithmetic sequence explicit formula, we get
a\(_n\) = a + (n - 1) d
a\(_{100}\) = (1/4)+ (100- 1) (1/4)=25.
