What should come in place of the question mark ‘?’ in the following number series?1680, 210, 30, 5, ?, 0.25

1. Break Down the Problem

We need to identify the pattern in the series: 1680, 210, 30, 5, ?, 0.25. Let's look at the relationship between the consecutive numbers.

2. Relevant Concepts

We will analyze the ratio or operation applied to each pair of consecutive numbers in the series to find the next term.

3. Analysis and Detail

• Start with $1680$ and move to $210$: $\frac{1680}{210} = 8 \implies 1680 \div 8 = 210$

• Next, from $210$ to $30$: $\frac{210}{30} = 7 \implies 210 \div 7 = 30$

• Then, from $30$ to $5$: $\frac{30}{5} = 6 \implies 30 \div 6 = 5$

Noticing the pattern, we see that each number is divided by a decreasing integer (8, 7, 6...).

• The next number should be: $5 \div 5 = 1$

• Finally, we verify the transition from $1$ to $0.25$: $1 \div 4 = 0.25$

4. Verify and Summarize

The pattern is consistent: each term is divided by an integer that decreases by 1 each time. Therefore, the numeral that replaces the question mark is confirmed.

Final Answer

The missing term in the series is $1$.