What Number Comes Next in the Sequence: 1 2 5 12 27 58?

Identifying the pattern in a sequence is a fascinating challenge that can enhance our problem-solving skills. In this article, we will explore a particular sequence and determine its next number. We'll employ a methodical approach to uncover the underlying structure and arrive at a logical conclusion. Let's dive in!

Understanding the Sequence

The given sequence is:

1, 2, 5, 12, 27, 58

To find the next number, we first need to analyze the differences between consecutive terms to identify any patterns.

First-Level Differences

Calculate the differences between consecutive terms:

2 - 1  15 - 2  312 - 5  727 - 12  1558 - 27  31

This gives us the first difference sequence:

1, 3, 7, 15, 31

Second-Level Differences

Calculate the differences of the first difference sequence:

3 - 1  27 - 3  415 - 7  831 - 15  16

This gives us the second difference sequence:

2, 4, 8, 16

Third-Level Differences

Calculate the differences of the second difference sequence:

4 - 2  28 - 4  416 - 8  8

This gives us the third difference sequence:

2, 4, 8

Recognize the Pattern

Notice the pattern in the differences:

2, 4, 8, 16 is a sequence of powers of 2 (2^1, 2^2, 2^3, 2^4). 

We can deduce that the next term in this sequence would be 32 (2^5).

Compute the Next Term in the Sequence

Add this to the last term of the first difference sequence:

31   32  63

Finally, add this to the last term of the original sequence:

58   63  121

Therefore, the next number in the sequence is 121.

Alternative Approach Using Quadratic Sequence Formula

To confirm our findings, we can consider a simpler description of the given sequence - an increasing quadratic sequence. The general term rule when n 1, 2, 3, ... is:

tn  2n^2 - 3n   3

To find the next term in the sequence, we use n 7:

t7  2(7^2) - 3(7)   3t7  2(49) - 21   3t7  98 - 21   3t7  80

Conclusion

We have arrived at 121 as the next number in the sequence, which is in line with our quadratic sequence approach, confirming the accuracy of our previous findings.

Related Keywords

sequence analysis pattern recognition quadratic sequence

Note: For more information on quadratic sequences, you may refer to search terms like 'quadratic sequence formula' and 'sequence solving methods.'