Introduction:
In mathematics, a sequence is a list of numbers that follow a specific pattern or rule. One important concept in sequences is the common difference. The common difference refers to the constant difference between consecutive terms in an arithmetic sequence. In simpler terms, it is the amount by which each term in a sequence increases or decreases.
To understand the concept of the common difference, it is essential to first understand what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers where each term after the first is obtained by adding a fixed number to the preceding term. This fixed number is known as the common difference, and it remains constant throughout the sequence. For example, the sequence 3, 6, 9, 12, 15, … is an arithmetic sequence with a common difference of 3.
In general, an arithmetic sequence can be represented as:
a, a + d, a + 2d, a + 3d, a + 4d, …
where a is the first term of the sequence, d is the common difference, and the ellipsis indicates that the sequence continues indefinitely. In this sequence, each term is obtained by adding d to the preceding term.
The common difference can be positive, negative, or zero. If the common difference is positive, the sequence increases from term to term. If the common difference is negative, the sequence decreases from term to term. If the common difference is zero, the sequence remains constant.
For example, the sequence 5, 10, 15, 20, 25, … is an arithmetic sequence with a common difference of 5. Each term in the sequence is obtained by adding 5 to the preceding term. On the other hand, the sequence 12, 8, 4, 0, -4, … is an arithmetic sequence with a common difference of -4. Each term in the sequence is obtained by subtracting 4 from the preceding term.
The common difference is an important concept in mathematics because it allows us to predict future terms in a sequence. By knowing the first term and the common difference of an arithmetic sequence, we can determine any term in the sequence. For example, if we know that the first term of an arithmetic sequence is 2 and the common difference is 3, we can determine that the fifth term in the sequence is 2 + 4(3) = 14.
In addition to predicting future terms, the common difference also allows us to find the sum of a finite arithmetic sequence. The sum of the first n terms of an arithmetic sequence can be calculated using the following formula:
S = n/2(2a + (n-1)d)
where S is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms. For example, the sum of the first 10 terms of the arithmetic sequence 2, 5, 8, 11, … can be calculated using the formula:
S = 10/2(2(2) + (10-1)(3)) = 10/2(4 + 27) = 155
Therefore, the sum of the first 10 terms of the sequence is 155.
The common difference is also useful in real-life situations. For example, if you receive a fixed amount of money each week, and you save a constant percentage of it, the amount of money you save each week will form an arithmetic sequence with a common difference equal to the amount you save. Knowing the common difference allows you to predict how much money you will save in the future.
Definition of Common Difference:
The common difference is a term used in arithmetic sequences. An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed value to the previous term. The fixed value is called the common difference. In other words, the common difference is the difference between any two consecutive terms in an arithmetic sequence.
The common difference is denoted by the letter “d.” It can be positive, negative, or zero. If the common difference is positive, it means that each term in the sequence is greater than the previous term. If the common difference is negative, it means that each term in the sequence is less than the previous term. If the common difference is zero, it means that all the terms in the sequence are the same.
Examples of Common Difference:
Example 1: Find the common difference in the arithmetic sequence 2, 4, 6, 8, 10.
Solution: To find the common difference, we subtract any two consecutive terms in the sequence. For example, we can subtract the second term from the first term:
4 – 2 = 2
Similarly, we can subtract the third term from the second term:
6 – 4 = 2
We can do the same for the other pairs of consecutive terms. We see that the difference is always 2. Therefore, the common difference is 2.
Example 2: Find the common difference in the arithmetic sequence -3, -1, 1, 3, 5.
Solution: To find the common difference, we subtract any two consecutive terms in the sequence. For example, we can subtract the second term from the first term:
-1 – (-3) = 2
Similarly, we can subtract the third term from the second term:
1 – (-1) = 2
We can do the same for the other pairs of consecutive terms. We see that the difference is always 2. Therefore, the common difference is 2.
Example 3: Find the common difference in the arithmetic sequence 10, 7, 4, 1, -2.
Solution: To find the common difference, we subtract any two consecutive terms in the sequence. For example, we can subtract the second term from the first term:
7 – 10 = -3
Similarly, we can subtract the third term from the second term:
4 – 7 = -3
We can do the same for the other pairs of consecutive terms. We see that the difference is always -3. Therefore, the common difference is -3.
Example 4: Find the common difference in the arithmetic sequence 1, 1, 1, 1, 1.
Solution: To find the common difference, we subtract any two consecutive terms in the sequence. For example, we can subtract the second term from the first term:
1 – 1 = 0
Similarly, we can subtract the third term from the second term:
1 – 1 = 0
We can do the same for the other pairs of consecutive terms. We see that the difference is always 0. Therefore, the common difference is 0.
Quiz
- What is the common difference of an arithmetic sequence?
- The common difference of an arithmetic sequence is the constant difference between any two consecutive terms in the sequence.
- Can the common difference of an arithmetic sequence be negative?
- Yes, the common difference of an arithmetic sequence can be negative.
- If the first term of an arithmetic sequence is 2 and the common difference is 3, what is the fifth term?
- The fifth term would be 14, since the sequence would be 2, 5, 8, 11, 14.
- What is the formula for finding the nth term of an arithmetic sequence?
- The formula is: a_n = a_1 + (n – 1)d, where a_n is the nth term, a_1 is the first term, and d is the common difference.
- What is the formula for finding the sum of the first n terms of an arithmetic sequence?
- The formula is: S_n = (n/2)(a_1 + a_n), where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the nth term.
- Can two arithmetic sequences have the same first term and different common differences?
- Yes, two arithmetic sequences can have the same first term and different common differences.
- Can two arithmetic sequences have the same common difference and different first terms?
- Yes, two arithmetic sequences can have the same common difference and different first terms.
- What is the common difference of the sequence 3, 6, 9, 12, …?
- The common difference is 3, since each term is 3 more than the previous term.
- What is the common difference of the sequence 10, 7, 4, 1, …?
- The common difference is -3, since each term is 3 less than the previous term.
- What is the 20th term of the sequence -2, 0, 2, 4, …?
- The 20th term would be 38, since the sequence would be -2, 0, 2, 4, …, 38. The formula a_n = a_1 + (n – 1)d can also be used to find this term: a_20 = -2 + (20 – 1)2 = 38.
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