Find the missing number in the series: 5, 4, 8, 9, 11, 14, 14 ?

To find the missing number in the series \(5, 4, 8, 9, 11, 14, 14, ?\), we need to look for a pattern. Let's examine the differences between consecutive terms:

1. \(4 - 5 = -1\)
2. \(8 - 4 = 4\)
3. \(9 - 8 = 1\)
4. \(11 - 9 = 2\)
5. \(14 - 11 = 3\)
6. \(14 - 14 = 0\)

We can see the differences are: \(-1, 4, 1, 2, 3, 0\).

Let's denote the differences:
\[
\begin{align*}
a_2 - a_1 &= -1 \\
a_3 - a_2 &= 4 \\
a_4 - a_3 &= 1 \\
a_5 - a_4 &= 2 \\
a_6 - a_5 &= 3 \\
a_7 - a_6 &= 0 \\
\end{align*}
\]

If we analyze the differences, we don't see an immediate arithmetic or geometric sequence, but the sequence of differences might suggest some pattern when summed cumulatively.

Let's apply the next difference to find the missing number \(a_8\):
\[
a_8 - a_7 = 1
\]

Thus,
\[
a_8 = a_7 + 1 = 14 + 1 = 15
\]

Therefore, the missing number in the series is \(15\).

Would you like more details or have any other questions?

Here are some relative questions to consider:
1. What are the common types of sequences and series in mathematics?
2. How do you identify patterns in a sequence?
3. What is the general form of an arithmetic sequence?
4. What is the general form of a geometric sequence?
5. How can you find the nth term of a sequence?
6. What are Fibonacci sequences and how do they differ from arithmetic and geometric sequences?
7. What is a recursive sequence?
8. How can differences between terms help identify the type of sequence?

**Tip:** When solving sequences, look for patterns in the differences or ratios between terms to identify the rule governing the sequence.

Discover how to find the missing number in the sequence 5, 4, 8, 9, 11, 14, 14 using pattern recognition and sequence analysis. Learn about arithmetic and geometric sequences and identify the rule governing this sequence. Suitable for students in Grades 9-12.

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