The problem asks to find the missing number in the sequence: 326, 452, 678, 1004, _____, 1956, 2582, 3308, 4134.

Identifying the Series Pattern

Let's examine the differences between consecutive terms in the series:

• $452 - 326 = 126$
• $678 - 452 = 226$
• $1004 - 678 = 326$

Notice that the differences themselves form a sequence: 126, 226, 326, ...

Analyzing the Differences Sequence

Now, let's find the differences between the terms in this new sequence of differences:

• $226 - 126 = 100$
• $326 - 226 = 100$

The difference between consecutive differences is constant and equals 100. This indicates an arithmetic progression in the differences.

Calculating the Missing Term

Following the pattern, the next difference should be $326 + 100 = 426$.

To find the missing term in the original series, add this difference to the preceding term (1004):

• $1004 + 426 = 1430$

Let's verify this by checking the subsequent terms.

• The difference after 426 should be $426 + 100 = 526$.
• The term following the missing term is 1956. Let's check: $1956 - 1430 = 526$. This matches our calculated difference.
• The next difference should be $526 + 100 = 626$.
• The next term is 2582. Let's check: $2582 - 1956 = 626$. This also matches.

The pattern is confirmed. The missing number is 1430.