A number series is given with one term missing. Choose the correct alternative from the options that will complete the series: 1, 5, 30, 210, 1680, ?

Introduction / Context:
This number series again uses a multiplicative pattern in which each term is produced by multiplying the previous one by a growing integer. The task is to identify the changing multipliers and extend that pattern to find the missing next term. Such structures are often inspired by factorial-like growth and are common in competitive reasoning exams.

Given Data / Assumptions:
Given series: 1, 5, 30, 210, 1680, ?We must determine the term that follows 1680.We assume a consistent pattern in the integer multipliers between consecutive terms.

Concept / Approach:
To solve this, we examine the ratio between each pair of adjacent terms. If these ratios follow a recognizable pattern, such as increasing consecutive integers, we can use that to find the next ratio and compute the missing term. When numbers increase very quickly, multiplication by larger and larger integers is usually involved rather than simple addition.

Step-by-Step Solution:
Step 1: Compute 5 / 1 = 5.Step 2: Compute 30 / 5 = 6.Step 3: Compute 210 / 30 = 7.Step 4: Compute 1680 / 210 = 8.Step 5: The multipliers are 5, 6, 7, 8, forming an increasing sequence of consecutive integers.Step 6: The next logical multiplier is 9.Step 7: Multiply 1680 by 9: 1680 * 9 = 15120, which gives the missing term.

Verification / Alternative check:
Write the extended series including the candidate term: 1, 5, 30, 210, 1680, 15120. Now check the multipliers: 1 * 5 = 5, 5 * 6 = 30, 30 * 7 = 210, 210 * 8 = 1680, 1680 * 9 = 15120. The pattern of multipliers 5, 6, 7, 8, 9 is perfectly consistent and typical of exam questions that build on consecutive integer factors. This confirms 15120 as the correct answer.

Why Other Options Are Wrong:
Option 13440 corresponds to 1680 * 8, which repeats the previous multiplier and breaks the increasing pattern.Option 11760 equals 1680 * 7, which would reverse the direction of the multipliers instead of continuing them forward.Option 8400 equals 1680 * 5, discarding the established pattern of steadily increasing multipliers from 5 to 9.

Common Pitfalls:
Some candidates try to approximate pattern recognition by averaging or by using arbitrary factors. Others may miscalculate products involving large numbers, such as 1680 * 9. It is important to systematically compute ratios, identify the clear integer sequence in the multipliers, and then perform the multiplication accurately to avoid such mistakes.

Final Answer:
The missing term that completes the product-based number series is 15120.