Find the next number in the series 18, 90, 450, 2250, ?

Introduction / Context:

This is a classic geometric style number series, where each term is obtained by multiplying the previous one by a constant factor. Such questions are fundamental in aptitude tests because they require recognition of simple multiplicative relationships. Once the common ratio is detected, predicting the next term is straightforward and involves only basic multiplication.

Given Data / Assumptions:

• The series given is 18, 90, 450, 2250, ?.
• We must find the single missing term that follows 2250.
• The numbers increase at a uniform multiplicative rate.
• There is no additional addition or subtraction term; the pattern is purely multiplicative.

Concept / Approach:

In geometric series questions, the central idea is the common ratio. We divide each term by its predecessor to see if the resulting ratio is constant. If the same ratio appears consistently, the series is geometric with that ratio. We then multiply the last known term by this ratio to obtain the next term. The steps are short and mechanical once the ratio is identified, making this type of question very scoring for well prepared candidates.

Step-by-Step Solution:

Step 1: Compute the ratio of the second term to the first: 90 / 18 = 5. Step 2: Compute the ratio of the third term to the second: 450 / 90 = 5, which matches the previous ratio. Step 3: Compute the ratio of the fourth term to the third: 2250 / 450 = 5, confirming a constant ratio of 5 throughout. Step 4: Since the series is geometric with common ratio 5, the next term is 2250 * 5. Step 5: Calculate 2250 * 5 = 11250. This is the required next term.

Verification / Alternative check:

Write the series explicitly with the ratio: 18, 18 * 5 = 90, 90 * 5 = 450, 450 * 5 = 2250, 2250 * 5 = 11250. Each step involves a clean multiplication by 5 with no extra modification. There are no deviations from this rule, and using 11250 as the next term keeps the series perfectly consistent. Any different value would create a break in the otherwise flawless geometric pattern.

Why Other Options Are Wrong:

Option 4520 is almost twice 2250, but the ratio 4520 / 2250 is not equal to 5 and fails the geometric rule. Option 6520 similarly gives a ratio that does not equal 5. Option 7250 also fails the constant ratio test, as 7250 / 2250 is not 5. Option 9000, though large, yields a ratio of 4, which contradicts the clearly established ratio of 5 among the earlier terms. None of these alternatives respect the simple multiply by 5 pattern.

Common Pitfalls:

Some students are tempted to look at differences between terms, but in geometric series the differences typically grow irregularly, which can be misleading. Others may mentally approximate the ratio and choose a nearby value instead of calculating precisely. Another error is to see large numbers and assume a more complex rule than necessary. The key is to always start by checking the ratio between consecutive terms for potential geometric behavior, especially when the jumps are large and regular looking as in this example.

Final Answer:

The next term in the geometric series with common ratio 5 is 11250.