In the number series 0.25, 8, 128, 1024, ?, find the next term by identifying the pattern in the powers of 2 used to generate the sequence.

Introduction / Context:
This question presents a number series that grows very rapidly: 0.25, 8, 128, 1024, ?. The presence of powers of 2 is a strong hint, since 8, 128 and 1024 are all very familiar powers of 2. The challenge is to uncover how the exponents change from term to term and then use that pattern to find the missing value.

Given Data / Assumptions:
The known terms of the series are:

• 0.25
• 8
• 128
• 1024
• ?

We assume that each term can be written as a power of 2, and that the exponents form a simple pattern that we can extend to find the next exponent and hence the next term.

Concept / Approach:
First express each term as 2 raised to a power. Then examine the sequence of exponents. If the differences between these exponents follow a recognisable pattern, such as a decreasing arithmetic progression, we can predict the next exponent and compute the corresponding power of 2.

Step-by-Step Solution:
Step 1: Rewrite each term as a power of 2.0.25 = 1/4 = 2^(-2).8 = 2^3.128 = 2^7.1024 = 2^10.Step 2: List the exponents: −2, 3, 7, 10.Step 3: Find the differences between exponents.From −2 to 3 the difference is +5.From 3 to 7 the difference is +4.From 7 to 10 the difference is +3.Step 4: The differences themselves follow a simple sequence: 5, 4, 3, which decreases by 1 each time. The next difference should therefore be +2.Step 5: Add this next difference to the last exponent: 10 + 2 = 12.Step 6: The next term must therefore be 2^12 = 4096.

Verification / Alternative check:
We can summarise the rule as: start with exponent −2, then successively add 5, 4, 3 and 2 to get the next exponents. This gives the sequence of exponents −2, 3, 7, 10, 12, which correspond to the numbers 0.25, 8, 128, 1024 and 4096. This reconstruction is fully consistent and leaves no ambiguity about the next term.

Why Other Options Are Wrong:
The other options 2526, 8486 and 2620 are not powers of 2 and do not fit naturally into the pattern of changing exponents. If we tried to replace 4096 with any of them, the neat stepwise pattern in the powers of 2 would break completely, so they cannot represent the correct continuation of the series.

Common Pitfalls:
Some learners may try to reason directly from the raw numbers and look at ratios like 8 / 0.25 or 128 / 8, which are not constant. The key is to notice that all visible terms except the first are familiar powers of 2. Once you convert 0.25 to 2^(-2), the exponent pattern becomes much clearer and the problem becomes straightforward.

Final Answer:
The next number in the series, following the pattern in the powers of 2, is 4096.