Number Series: Finding the Missing Term

This question asks us to identify the missing number in a given sequence: 14, 158, 258, 322, ?, 374. To find the missing term, we need to analyze the pattern or rule governing the sequence.

Analyzing the Series Pattern

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

• The difference between the second and first term is: $158 - 14 = 144$.
• The difference between the third and second term is: $258 - 158 = 100$.
• The difference between the fourth and third term is: $322 - 258 = 64$.

We observe the differences are 144, 100, and 64. Let's see if these numbers have a recognizable pattern. We can see that these are perfect squares:

• $144 = 12^2$
• $100 = 10^2$
• $64 = 8^2$

The pattern in the differences is a sequence of decreasing squares: $12^2, 10^2, 8^2$. The base numbers are decreasing by 2 each time (12, 10, 8).

Calculating the Missing Number

Following this pattern, the next difference should be the square of the next number in the sequence (8 - 2 = 6), which is $6^2$.

• Calculate the next difference: $6^2 = 36$.

Now, we add this difference to the last known term before the question mark (322) to find the missing term:

• Missing term = $322 + 36 = 358$.

Verifying the Pattern

To ensure our pattern is correct, let's check if the subsequent term (374) fits the pattern. The difference after $6^2$ should be $4^2$ (since 6 - 2 = 4).

• Calculate the next difference: $4^2 = 16$.
• Add this difference to our calculated missing term: $358 + 16 = 374$.

This matches the last term given in the series (374). Therefore, the pattern is confirmed.

Conclusion

The number series follows the rule of adding decreasing squares ($12^2, 10^2, 8^2, 6^2, 4^2, \dots$). The missing number in the series 14, 158, 258, 322, ?, 374 is 358.