In the number series 6, 6, 9, 18, 45, ?, find the next term by analysing the pattern in the changing multiplication factors between terms.

Introduction / Context:
This problem gives a series 6, 6, 9, 18, 45, ? and asks you to find the next number. The terms increase, but the growth is not purely arithmetic or geometric. Instead, the multipliers between consecutive terms themselves follow a simple pattern, which is the key to solving the question.

Given Data / Assumptions:
The known terms are:

• 6
• 6
• 9
• 18
• 45
• ?

We assume that each term (from the second onward) is obtained by multiplying the previous term by a factor that gradually increases in a regular way.

Concept / Approach:
To reveal the pattern, we calculate the ratio of each term to the previous term. If these ratios form a simple sequence, such as starting at 1 and increasing by 0.5 each time, we can extend that sequence to find the next ratio and thus the next term in the series.

Step-by-Step Solution:
Step 1: Find the ratio from the first term to the second: 6 / 6 = 1. So the second term is 6 * 1.Step 2: Find the ratio from the second term to the third: 9 / 6 = 1.5.Step 3: Find the ratio from the third term to the fourth: 18 / 9 = 2.Step 4: Find the ratio from the fourth term to the fifth: 45 / 18 = 2.5.Step 5: The ratios are 1, 1.5, 2 and 2.5. These increase by 0.5 each time.Step 6: Following this pattern, the next ratio should be 2.5 + 0.5 = 3.Step 7: Multiply the last known term by this ratio to get the next term: 45 * 3 = 135.

Verification / Alternative check:
We can restate the rule as follows: Start at 6, then multiply by 1 to get 6, by 1.5 to get 9, by 2 to get 18, by 2.5 to get 45 and by 3 to get 135. The pattern in the multipliers is extremely regular: 1, 1.5, 2, 2.5, 3, which confirms that the series is built in a systematic way and that 135 is the only value that fits this structure.

Why Other Options Are Wrong:
Values 87, 99.75 and 115.5 do not equal 45 multiplied by 3 and do not fit any simple extension of the ratio pattern. If any of them were used, the multipliers between terms would no longer increase by a consistent 0.5 step, breaking the logical progression of the sequence.

Common Pitfalls:
Some learners focus exclusively on differences between consecutive terms instead of looking at ratios. In series where the magnitudes of terms grow in a way that suggests repeated multiplying rather than adding, checking the ratios is a superior strategy that often reveals a clear pattern.

Final Answer:
The next term in the series, obtained by multiplying 45 by 3, is 135.