True or False
1. – 5, – 5, – 5, – 5, – 5, … is an arithmetic sequence.
2. In an arithmetic sequence, it is possible that the 13th term is equal to its 53rd term.
3. In an arithmetic sequence, the common difference is computed by subtracting the first term from the second term even if the common difference is not zero.
4. The terms in an arithmetic sequence are always arranged in an ascending order.
5. The series of the arithmetic sequence 1, 2, 3, 4, 5, … is 15.
My answer are all true except for number 4 which is false. Am I right?
1. False. The given sequence, -5, -5, -5, -5, -5, ..., is a constant sequence, not an arithmetic sequence, because there is no common difference between terms.
2. True. In an arithmetic sequence, it is possible for the 13th term to be equal to the 53rd term. This would only occur if the common difference is zero, and all terms in the sequence would be the same.
3. False. In an arithmetic sequence, the common difference is computed by subtracting the second term from the first term, not the other way around. The common difference is found by subtracting the previous term from the current term.
4. False. The terms in an arithmetic sequence can be arranged in any order, whether it is ascending, descending, or mixed.
5. False. The sum of an arithmetic sequence is calculated using the formula S = (n/2)(2a + (n-1)d), where n is the number of terms, a is the first term, and d is the common difference. For the given sequence 1, 2, 3, 4, 5, ..., if we substitute n = 5, a = 1, and d = 1 into the formula, we get S = (5/2)(2(1) + (5-1)(1)) = 15. So, the sum of the sequence is 15, not the series.
1. False: An arithmetic sequence has a constant difference between consecutive terms. In this case, the terms are all -5, so the difference between consecutive terms is zero.
2. True: In an arithmetic sequence, it is possible for the 13th term to be equal to its 53rd term. This can happen when the common difference is zero or when the sequence has a repeating pattern.
3. False: The common difference in an arithmetic sequence is determined by subtracting the second term from the first term, not the other way around. By subtracting the first term from the second term, you can find the common difference.
4. False: The terms in an arithmetic sequence can be arranged in ascending or descending order, depending on the direction of the sequence. An ascending sequence would have increasing terms, while a descending sequence would have decreasing terms.
5. False: The given arithmetic sequence 1, 2, 3, 4, 5, … is an example of an arithmetic progression with a common difference of 1. The series of this sequence can be found using the formula for the sum of an arithmetic series, which is Sn = (n/2)(a + l), where Sn is the sum, n is the number of terms, a is the first term, and l is the last term. For this series, the last term is infinite, so there is no finite sum (it diverges to positive infinity).