Insert terms in the arithmetic sequence:
1. 17 and 5
2. 4 and 24
3. -1, 12, 16
4. 1, 3, 1, -1, -1, -17
Answer (2)
For the sequence 17 and 5, insert 11 to form the arithmetic sequence 17, 11, 5.
For the sequence 4 and 24, insert 14 to form the arithmetic sequence 4, 14, 24.
It is not possible to determine the terms to insert for sequences 3 and 4 without more information.
The inserted terms for sequences 1 and 2 are 11 , 14 .
Explanation
Understanding the Problem We are given four sequences and asked to insert terms to make them arithmetic sequences. An arithmetic sequence is a sequence where the difference between consecutive terms is constant. We will analyze each sequence separately.
Sequence 1: Inserting one term
The sequence is 17 and 5. We want to insert one term, x , between them to form an arithmetic sequence: 17, x , 5. In an arithmetic sequence, the difference between consecutive terms is constant. Therefore, x − 17 = 5 − x . Solving for x , we get 2 x = 22 , so x = 11 . The common difference is 11 − 17 = − 6 . The sequence is 17, 11, 5.
Sequence 2: Inserting one term
The sequence is 4 and 24. We want to insert one term, x , between them to form an arithmetic sequence: 4, x , 24. In an arithmetic sequence, the difference between consecutive terms is constant. Therefore, x − 4 = 24 − x . Solving for x , we get 2 x = 28 , so x = 14 . The common difference is 14 − 4 = 10 . The sequence is 4, 14, 24.
Sequence 3: Unclear Insertion
The sequence is -1, -1, 12, 16. We need to determine how many terms to insert to make it an arithmetic sequence. The difference between -1 and -1 is 0. The difference between -1 and 12 is 13. The difference between 12 and 16 is 4. This is not an arithmetic sequence. It is unclear how many terms to insert to make it an arithmetic sequence. Let's assume we want to insert terms between -1 and 12, and between 12 and 16 such that the entire sequence becomes arithmetic. This is not possible to determine without more information.
Sequence 4: Unclear Insertion
The sequence is 1, 3, 1, -1, -1, -17. We need to determine how many terms to insert to make it an arithmetic sequence. The difference between 1 and 3 is 2. The difference between 3 and 1 is -2. The difference between 1 and -1 is -2. The difference between -1 and -1 is 0. The difference between -1 and -17 is -16. This is not an arithmetic sequence. It is unclear how many terms to insert to make it an arithmetic sequence. This is not possible to determine without more information.
Conclusion In summary, for sequences 1 and 2, we found the missing terms to create arithmetic sequences. For sequences 3 and 4, it is not possible to determine the number of terms to insert to make them arithmetic sequences without further information.
Examples
Arithmetic sequences are used in various real-life scenarios, such as calculating simple interest, predicting the number of seats in rows of a theater, and determining patterns in evenly spaced objects. Understanding how to insert terms into a sequence to make it arithmetic helps in predicting future values and maintaining consistent patterns in these scenarios. For example, if you are saving money each month with a constant increase, the amounts saved form an arithmetic sequence.
For the sequences given, we can insert 11 between 17 and 5, and 14 between 4 and 24 to form valid arithmetic sequences. The sequences become 17, 11, 5 and 4, 14, 24 respectively. However, for sequences 3 and 4, we cannot determine how to insert terms to make them arithmetic without more information.
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