Number Series Analysis

The given number series is $45, 40, 50, ?, 55, 30$. To find the missing number, we need to identify the pattern governing the series. Upon closer inspection, the series appears to be composed of two interleaved arithmetic progressions.

Interleaved Series Identification

Let's separate the series into two based on the position of the numbers:

• Odd Positions (1st, 3rd, 5th): $45, 50, 55$
• Even Positions (2nd, 4th, 6th): $40, ?, 30$

Pattern in Odd Positions

The series in odd positions ($45, 50, 55$) increases by a constant difference:

• $50 - 45 = 5$
• $55 - 50 = 5$

This is an arithmetic progression with a common difference of $5$.

Pattern in Even Positions

The series in even positions ($40, ?, 30$) should also follow a consistent pattern, likely an arithmetic progression.

Let the missing number be $x$. The sequence is $40, x, 30$.

If this is an arithmetic progression, let the common difference be $d$. Then:

• $x = 40 + d$
• $30 = x + d$

Substitute the first equation into the second:

$30 = (40 + d) + d$

$30 = 40 + 2d$

Now, solve for $d$:

$2d = 30 - 40$

$2d = -10$

$d = \frac{-10}{2}$

$d = -5$

Calculating the Missing Number

Using the common difference $d = -5$, we can find the missing number $x$ (which is the 4th term in the original series):

$x = 40 + d$

$x = 40 + (-5)$

$x = 35$

The complete series is $45, 40, 50, 35, 55, 30$.