Sequence Pattern Analysis

The problem asks to find a possible value for '$x$' in the given number sequence: 6, 9, 14, $x$, 30, 41.

We need to identify the pattern governing the sequence.

Identifying the Pattern

Let's examine the differences between consecutive terms:

• $9 - 6 = 3$
• $14 - 9 = 5$
• Let the difference between $x$ and 14 be $d_1$. So, $x - 14 = d_1$.
• Let the difference between 30 and $x$ be $d_2$. So, $30 - x = d_2$.
• $41 - 30 = 11$

The sequence of differences starts with 3, 5, ... , 11.

Calculating the Missing Term '$x$'

Observe the pattern in the differences: 3, 5. It appears the difference increases by 2 each time.

• The difference after 5 should be $5 + 2 = 7$.
• Therefore, $d_1 = 7$.
• Using $x - 14 = d_1$, we get $x - 14 = 7$.
• Solving for $x$: $x = 14 + 7 = 21$.

Let's verify this value using the next difference:

• The difference after $d_1$ (which is 7) should be $7 + 2 = 9$.
• So, $d_2$ should be 9.
• Let's check if $30 - x = 9$ when $x = 21$.
• $30 - 21 = 9$. This matches the expected pattern.

The complete sequence of differences is 3, 5, 7, 9, 11, which follows a consistent pattern of adding 2.

Thus, the possible value of $x$ is 21.