What Is The Next Number In This Sequence Of Numbers? ✓ Solved

Given this sequence of numbers: a) What is the next number?

Given the sequence of numbers: 9, 13, 19, 27, we need to find the next number in the sequence and also create an algorithm that can solve this problem for any sequence of numbers.

To identify the next number, let’s start by examining the differences between consecutive terms in the given sequence. The calculations are as follows:

  • 13 - 9 = 4
  • 19 - 13 = 6
  • 27 - 19 = 8

The differences are 4, 6, and 8. We can observe that the differences themselves increase by 2 each time. This means we can formulate this as a second sequence based on the first differences:

Given the differences:

  • The first difference = 4
  • The second difference = 6
  • The third difference = 8

If we continue this pattern, the next difference should be:

  • Next difference = 8 + 2 = 10

Now, to find the next number in the original sequence:

  • Next number = Last number (27) + Next difference (10) = 27 + 10 = 37

Thus, the next number in the sequence is 37.

Algorithm to Solve This Problem for Any Numbers

1. Begin with a sequence of numbers.

2. Compute the first difference by subtracting each consecutive number in the sequence.

3. Analyze the pattern of the first differences to determine if they create a sequence, such as linear, quadratic, or higher order differences.

4. If a pattern in the first differences is identified, apply it to find the next difference.

5. Add this new difference to the last number in the original sequence to find the next number.

6. For any given n value in the original sequence, use the derived formula to compute S.

Finding S for n = 0 and n = 10

Now let’s evaluate S values at n = 0 and n = 10 in the sequence given: n=1 S=9, n=2 S=13, n=3 S=19, n=4 S=27.

From our earlier analysis, we determined that the relationship between n and S can be modeled through a simple equation derived from the series of numbers: S = 9 + (n - 1) 4 + 2 ((n - 1) * (n - 2)) / 2.

To find S for n = 0:

  • If n = 0, following our pattern, we can derive:
  • S = 9 + (0 - 1) 4 + 2 ((0 - 1) * (0 - 2)) / 2
  • S = 9 - 4 + 2 (0) = 5

To find S for n = 10:

  • If n = 10, utilizing our formula:
  • S = 9 + (10 - 1) 4 + 2 ((10 - 1) * (10 - 2)) / 2
  • S = 9 + 9 4 + 2 (9 * 8) / 2
  • S = 9 + 36 + 72 = 117

Thus, we have:

  • S for n = 0 is 5
  • S for n = 10 is 117

References

  • Knuth, D. E. (1997). The Art of Computer Programming. Volume 1: Fundamental Algorithms.
  • Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to Algorithms.
  • Gibbons, A. (1985). Algorithmic Graph Theory and Perfect Graphs.
  • Sedgewick, R., & Wayne, K. (2011). Algorithms (4th Edition).
  • Goodrich, M. T., & Tamassia, R. (2014). Data Structures and Algorithms in Java.
  • Wirth, N. (1976). Algorithms + Data Structures = Programs.
  • Papadimitriou, C. H. (1994). Computational Complexity.
  • Aho, A. V., Hopcroft, J. E., & Ullman, J. D. (1974). The Design and Analysis of Computer Algorithms.
  • Kleinberg, J., & Tardos, É. (2005). Algorithm Design.
  • Cormen, T. H., & Leiserson, C. E. (1985). Introduction to Algorithms.

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