A sequence is very important in mathematical applications. It represents a large amount of data expressed in a specific order, called progressions, or a set containing different terms. The length of the sequence is defined as the number of elements present in a finite sequence. Furthermore, these elements may form a series. Strictly speaking, series and sequences are different terms. There are many popular sequences, such as Fibonacci sequence, infinite sequence, finite sequence, geometric sequence, etc. Out of those, one is an arithmetic sequence.
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What is Arithmetic Sequence?
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A sequence is a set that contains natural numbers. An arithmetic sequence also contains natural numbers who’s each term is created by subtracting or adding a common number to its preceding or succeeding term. Therefore, in an arithmetic sequence, the difference between its adjacent terms is the same.
The major property of an arithmetic sequence is if a constant is added, subtracted, multiplied by a constant, or divided by a non-zero number to each term in an arithmetic sequence, then the resulting sequence will also be an arithmetic sequence.
The video below explains this:
Arithmetic Progression Detailed Video Explanation:
Also Read : Properties of Arithmetic Progression
Formulas of Arithmetic Sequence
Mathematically, if a1, a2, a3 … are the terms of an arithmetic sequence, then,
Formula 1: an+1 = an + d
where, n = set of natural numbers
d = common difference
General expression of arithmetic sequence = a, a + d, a + 2d, a + 3d …
The general term, i.e., nth term in an arithmetic sequence is given by:
Formula 2: an = a + (n – 1) d
Read More: Sequence and Series
Last term in arithmetic sequence is denoted by l, and is given by:
Formula 3: l = a + (n – 1) d
The sum of terms in an arithmetic sequence is given by:
Formula 4: Sn = n [2a + (n – 1) d] / 2
We can write the above formula as,
Formula 5: Sn = n (a + l) / 2
Applications of Arithmetic Sequence Formula
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The arithmetic sequence formula is used in everyday life. Some of the common examples are:
- Stacking chairs, bowls, cups, pack of cards.
- Seats in an auditorium or stadium are arranged in an arithmetic sequence.
- The hands of a clock move in an arithmetic sequence.
- Weeks, years, and leap years’ work on arithmetic sequence formulas.
- Your age increases each year in an arithmetic sequence.
Check Important Notes for Permutation and Combination
Solved Examples of Arithmetic Sequence
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Example 1: What is the 8th term in the arithmetic sequence 4, 6, 8, 10 …?
Solution: In the given arithmetic sequence, we have,
a = 4
d = 6 – 4 = 2
We know, an = a + (n – 1) d
where, n = the term for which we want the value
Here, n = 8
Therefore, a8 = 4 + (8 – 1) 2
a8 = 18
This implies the 8th term in the arithmetic sequence will be 18.
Example 2: What is the 7th term in the arithmetic sequence 20, 18, 16 …?
Solution: In the given arithmetic sequence, we have,
a = 20
d = 18 – 20 = -2
We know, an = a + (n – 1) d
where, n = the term for which we want the value
Here, n = 7
Therefore, a8 = 20 + (7 – 1) (-2)
a8 = 8
This implies the 8th term in the given arithmetic sequence will be 8.
Example 3: What is the sum of terms in the series 7, 15, 23, 31 … 215?
Solution: Here, a = 7
d = 15 – 7 = 8
We know, l = a + (n – 1) d
215 = 7 + (n – 1) 8 = 7 + 8n - 8
Therefore, n = 27
Also, sum of terms = Sn = n (a + l) / 2
Sn = 27 (7 + 215) / 2 = 2997
Example 4: What will be the pth term in an arithmetic sequence, if mth term is n and the nth term is m, where m ≠ n?
Solution: From the given data, we have,
am = a + (m – 1) d = n ... (1)
And an = a + (n – 1) d = m ... (2)
Solving (1) and (2), we get,
(m – n) d = n – m,
d = – 1 ... (3)
And a = n + m – 1 ... (4)
Therefore ap= a + (p – 1) d = n + m – 1 + (p – 1) (–1) = n + m – p
Hence, the pth term is n + m – p.
Points to Remember
Following are some important points:
- There are many popular sequences which includes Fibonacci sequence, infinite sequence, finite sequence, geometric sequence. Arithmetic Sequence is also a part of it.
- Set of Natural Numbers is known as Arithmetic Sequence.
- An Arithmetic Sequence contains those natural numbers who’s each term is created by subtracting or adding a common number to its preceding or succeeding term.
- Main Formula is an+1 = an + d, where ‘n’ is set of natural numbers and ‘d’ is common difference.
Sample Questions
Ques: In the first year, the income of a businesswoman is Rs. 3, 00, 000. She receives an increment of Rs.10,000 per year for the next 19 years. What is the final amount she will receive in the next 20 years? (2 Marks)
Ans: Here, we have an A.P. with a = 3, 00, 000, d = 10,000, and n = 20.
Using the sum formula, we get,
S20 =20 [600000 + 19 x 10000] / 2 = 10 (790000) = 79, 00, 000.
Ques: What is the sum of the first 20 terms in the sequence 5, 10, 15, 20 …? (3 Marks)
Ans: We know, the sum of terms in an arithmetic sequence is given by,
Sn = n [2a + (n – 1) d] / 2
Here, a = 5
n = 20
d = 10 – 5 = 5
Therefore, we have,
Sn = 20 [2x 5 + (20 – 1) 5] / 2 = 1950.
Ques: What is the first in an arithmetic sequence with 10th term = 680 and its common difference = 24? (2 Marks)
Ans: We know, an = a + (n – 1) d
680 = a + (10 – 1) 24
On solving we get,
a = 464
Ques: What is the 100th term in arithmetic sequence if its 5th term is 22 and 15th term is 62? (3 Marks)
Ans: We know, an = a + (n – 1) d
From the data we have,
a5 = a + (5 – 1) d = 22
a15 = a + (15 – 1) d = 62
On solving these equations for a and d, we get,
a = 6, and d = 4
Now, the 100th term = a100 = a + (n – 1) d
a100 = 6 + (100 – 1) 4 = 402
Ques: What is the sum of all positive integers divisible by 6 from 6 to 1836? (3 Marks)
Ans: From the data, the series = 6, 12, 18 … 1836
Here, a = 6
d = 6
We know, l = a + (n – 1) d
1836 = 6 + (n – 1) 6 = 6 + 6n - 6
Therefore, n = 306
Also, sum of terms = Sn = n (a + l) / 2
Sn = 306 (6 + 1836) / 2 = 2, 81, 826
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