Arithmetic Sequences
An arithmetic sequence is a sequence in which the difference between each consecutive term is constant. An arithmetic sequence can be defined by an explicit formula in which an = d (n – 1) + c, where d is the common difference between consecutive terms, and c = a1. An arithmetic sequence can also be defined recursively by the formulas a1 = c, an+1 = an + d, in which d is again the common difference between consecutive terms, and c is a constant.
The sum of an infinite arithmetic sequence is either ∞, if d > 0, or – ∞, if d < 0.
There are two ways to find the sum of a finite arithmetic sequence. To use the first method, you must know the value of the first term a1 and the value of the last term an. Then, the sum of the first n terms of the arithmetic sequence is Sn = n(a1+an/2). To use the second method, you must know the value of the first term a1 and the common difference d. Then, the sum of the first n terms of an arithmetic sequence is Sn = na1 + n/2(dn – d ).
How explicit formulas work
Here is an explicit formula of the sequence 3, 5, 7,…
a(n)=3+2(n−1)
In the formula, n is any term number and a(n) is the nth term.
This formula allows us to simply plug in the number of the term we are interested in, and we will get the value of that term.
In order to find the fifth term, for example, we need to plug n=5 into the explicit formula.
Cool! This is in fact the fifth term of 3, 5, 7,…
Writing explicit formulas
Consider the arithmetic sequence 5,8,11,…The first term of the sequence is 3
We can get any term in the sequence by taking the first term 5 and adding the common difference 3 to it repeatedly. Check out, for example, the following calculations of the first few terms.
The table shows that we can get the nth term (where n is any term number) by taking the first term 5 and adding the common difference 3 repeatedly for n-1 times. This can be written algebraically as 5+3(n-1).
In general, this is the standard explicit formula of an arithmetic sequence whose first term is A and common difference is B:
A+B(n−1)
Equivalent explicit formulas
Explicit formulas can come in many forms.For example, the following are all explicit formulas for the sequence 3,5,7,…
A common misconception
An arithmetic sequence may have different equivalent formulas, but it’s important to remember that only the standard form gives us the first term and the common difference.
The explicit formula 2+6(n−1) describes this sequence, but the explicit formula 2+6n describes a different sequence.
In order to bring the formula 2+6(n−1) to an equivalent formula of the form A+Bn, we can expand the parentheses and simplify:
2+6(n−1)
=2+6n−6
=-4+6n
Some people might prefer the formula −4+6n over the equivalent formula 2+6(n−1), because it’s shorter. The nice thing about the longer formula is that it gives us the first term.
Using Explicit Formulas for Arithmetic Sequences
We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.
an=a1+d(n−1)
To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.
The common difference is −50 , so the sequence represents a linear function with a slope of −50 . To find the yy -intercept, we subtract −50 from 200:200−(−50)=200+50=250 . You can also find the yy -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis.
Recall the slope-intercept form of a line is y=mx+b. When dealing with sequences, we use anan in place of y and n in place of x. If we know the slope and vertical intercept of the function, we can substitute them for m and b in the slope-intercept form of a line. Substituting −50 for the slope and 250 for the vertical intercept, we get the following equation:
an=−50n+250
We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is an=200−50(n−1) , which simplifies to an=−50n+250.
A GENERAL NOTE: EXPLICIT FORMULA FOR AN ARITHMETIC SEQUENCE
An explicit formula for the nthnth term of an arithmetic sequence is given by
an=a1+d(n−1)
HOW TO: GIVEN THE FIRST SEVERAL TERMS FOR AN ARITHMETIC SEQUENCE, WRITE AN EXPLICIT FORMULA.
- Find the common difference, a2−a1.
- Substitute the common difference and the first term into an=a1+d(n−1)
EXAMPLE: WRITING THE NTH TERM EXPLICIT FORMULA FOR AN ARITHMETIC SEQUENCE
Write an explicit formula for the arithmetic sequence.
{2, 12, 22, 32, 42, …}
Show Solution
The common difference can be found by subtracting the first term from the second term.
d=a2−a1
=12−2
=10
The common difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify.
an=2+10(n−1)
an=10n−8
Analysis of the Solution
The graph of this sequence shows a slope of 10 and a vertical intercept of −8.
Sequences as Functions – Explicit
What Is Arithmetic Sequence Explicit Formula?
The arithmetic sequence explicit formula is used to find any term (nth term) of the arithmetic sequence, a1,a2,a3,…,an,…. using its first term (a) and the common difference (d). This formula gives the nth term formula of an arithmetic sequence. The arithmetic sequence explicit formula is:
Examples Using Arithmetic Sequence Explicit Formula
Example 1: Find the 15th term 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 15th term of the given sequence is calculated using,
an = a + (n – 1) d
a15 = -3 + (15 – 1) 2 = -3 + 14(2) = -3 + 28 = 25.
Answer: The 15th term of the given sequence = 25.
Example 2: Find the common difference of the arithmetic sequence whose first term is 1/2 and whose 10th term is 9.
Solution:
The first term is, a = 1/2.
Its 10th term is a10 = 9.
Using the arithmetic sequence explicit formula,
an = a + (n – 1) d
Substituting n = 10, we get
a10=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) nth term 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) nth term of an arithmetic sequence using the arithmetic sequence explicit formula.
an = a + (n – 1) d
an = -1/2 + (n – 1) (5/2)
an = – 1/2 + 5/2 n – 5/2
an = (5/2) n – 3
Answer: The nth term of the given sequence = (5/2) n – 3.
FAQs on Arithmetic Sequence Explicit Formula
What Is Arithmetic Sequence Explicit Formula?
The arithmetic sequence explicit formula is a formula that is used to find the nth term of an arithmetic sequence without computing any other terms before the nth term. Using this formula, the nth term of an arithmetic sequence whose first term is ‘a’ and common difference is ‘d’ is, an = a + (n – 1) d.
How To Derive Arithmetic Sequence Explicit Formula?
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 1st term is a = a + (1 – 1) d, the 2nd term is a + d = a + (2 – 1) d, and the 3rd term is a + 2d = a + (3 – 1) d. In the same way, the nth term is, ann = a + (n – 1) d.
What Are the Applications of Arithmetic Sequence Explicit Formula?
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 50th term 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 formula an = a + (n – 1) d, then we get a50 = (-7) + (50 – 1) 2 = 91.
What Is the 100th term of the Arithmetic Sequence 1/4, 1/2, 3/4, … Using Arithmetic Sequence Explicit Formula?
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
an = a + (n – 1) d
a100 = (1/4) + (100 – 1) (1/4) = 25.
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