Identifying the Next Term in a Mathematical Sequence
Welcome to our blog where we dive into the fascinating world of mathematical sequences. Today, we will explore a specific sequence and uncover the pattern underlying it. The sequence in question is a series of terms where each subsequent term is derived from the previous ones in an intriguing way. Let's uncover how to find the next term in the sequence 10, 10, 20, 35, 55, 80, and beyond!
Understanding the Sequence
The sequence provided is:
10 10 20 20 15 35 35 20 55 55 25 80From this, we can see that the sequence starts at 10 and then each next term is derived by adding a value that increases by 5 each time. Looking into this pattern, we can identify that the sequence is:
10, 10 20 20, 15 35 35, 20 55 55, 25 80 80, 30 110The Mathematical Formula
More formally, the sequence can be expressed using an arithmetic progression (AP). Let's denote the nth term of the sequence as an. We can write the sequence as:
S 10, 20, 35, 55, 80, ...
The pattern can be explained by the following differences between consecutive terms:
1 55 102 1010 203 2015 354 3520 555 5525 806 8030 110 [ANSWER]
We can observe that the difference between the terms follows an arithmetic progression with a common difference of 5, starting from 10:
D1 0 (initial difference) D2 10 D3 15 D4 20 D5 25 D6 30 (next difference)Using the formula for the nth term of an AP, where the common difference is 5, we can find the next term in the sequence. The nth term can be calculated as:
an an-1 5(n - 2)
Let's apply this to find the 6th term (80, 30 110) and the 7th term (110, 35 145).
Conclusion
Understanding and identifying patterns in sequences is a fundamental skill in mathematics. Whether it's an arithmetic progression or a more complex series, recognizing the underlying pattern can help us predict the next term. For the sequence 10, 10, 20, 35, 55, 80, the next term is 110, as calculated from the increasing differences of 5.
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