Series Pattern Identification

Calculate the differences between consecutive terms in the number series:

• $1248 - 887 = 361$
• $887 - 598 = 289$
• $598 - 429 = 169$

Identify these differences as perfect squares:

• $361 = 19^2$
• $289 = 17^2$
• $169 = 13^2$

Observe the sequence of the bases of these squares: 19, 17, 13. The differences between these bases are $19-17=2$ and $17-13=4$. This suggests an alternating pattern of differences: 2, 4, 2, 4, ...

Finding the Missing Number

Apply the identified pattern to find the next term:

• Following the pattern $13 \xrightarrow{-2} 11$, the next base is 11.
• Calculate the square: $11^2 = 121$.
• Subtract this square from the preceding term (429) to find the missing number: $429 - 121 = 308$.

The missing number in the series is 308.

Verifying Series Logic

Confirm the pattern using the calculated missing number (308) and the subsequent terms:

• The next base difference in the pattern is 4: $11 \xrightarrow{-4} 7$. The square is $7^2 = 49$.
• Check the subtraction: $308 - 49 = 259$. This matches the next term.
• The final base difference in the pattern is 2: $7 \xrightarrow{-2} 5$. The square is $5^2 = 25$.
• Check the subtraction: $259 - 25 = 234$. This matches the final term.

The pattern of subtracting decreasing squares ($19^2, 17^2, 13^2, 11^2, 7^2, 5^2$) is confirmed.