Number Series Analysis: Finding the Missing Term

The question asks us to identify the pattern in the given number series and find the missing number represented by the question mark (?). The series is:

$3, 3, 4.5, 9, 22.5, ?$

Identifying the Series Pattern

To find the missing number, we need to determine the relationship between consecutive terms in the series. Let's examine the ratio between each term and the term preceding it:

• Term 1 to Term 2: $3 \rightarrow 3$. The ratio is $\frac{3}{3} = 1$.
• Term 2 to Term 3: $3 \rightarrow 4.5$. The ratio is $\frac{4.5}{3} = 1.5$.
• Term 3 to Term 4: $4.5 \rightarrow 9$. The ratio is $\frac{9}{4.5} = 2$.
• Term 4 to Term 5: $9 \rightarrow 22.5$. The ratio is $\frac{22.5}{9} = 2.5$.

Calculating the Next Term

Observe the pattern in the ratios (multipliers): $1, 1.5, 2, 2.5$. We can see that each multiplier increases by $0.5$ compared to the previous one.

This suggests an arithmetic progression in the multipliers.

Therefore, the next multiplier in the sequence will be:

$2.5 + 0.5 = 3$

To find the missing number (the 6th term), we multiply the 5th term ($22.5$) by this next multiplier ($3$):

Missing Term = $22.5 \times 3$

Missing Term = $67.5$

Conclusion

The pattern involves multiplying the current term by an increasing factor, which grows by $0.5$ each step ($1, 1.5, 2, 2.5, 3, \dots$). Following this pattern, the missing number in the series is $67.5$.