A number series is given with one term missing. Choose the correct alternative from the given options that will complete the series: 1, 7, 42, 210, 840, ?

Introduction / Context:
This is a multiplicative number series where each term is obtained by multiplying the previous term by a whole number. Interestingly, the multipliers themselves follow a decreasing pattern. Such questions test the ability to recognize factorial-like or product-based structures in sequences, which are common in logical reasoning and quantitative aptitude exams.

Given Data / Assumptions:
Given series: 1, 7, 42, 210, 840, ?We need to find the next term after 840.We assume that each term is a product of the previous term and a systematically changing multiplier.

Concept / Approach:
The strategy is to find the ratio between each pair of consecutive terms. If these ratios form a recognizable pattern, such as descending integers, we can extend that pattern to determine the next ratio and thus compute the missing term. This method is especially effective when the series grows quickly, indicating a multiplicative rather than additive structure.

Step-by-Step Solution:
Step 1: Compute the ratio 7 / 1 = 7.Step 2: Compute the ratio 42 / 7 = 6.Step 3: Compute the ratio 210 / 42 = 5.Step 4: Compute the ratio 840 / 210 = 4.Step 5: The multipliers are 7, 6, 5, 4, which form a descending sequence of integers.Step 6: The next multiplier should therefore be 3.Step 7: Multiply 840 by 3 to get the next term: 840 * 3 = 2520.

Verification / Alternative check:
Write the extended series with the candidate term: 1, 7, 42, 210, 840, 2520. Now list the multipliers: 1 * 7 = 7, 7 * 6 = 42, 42 * 5 = 210, 210 * 4 = 840, and 840 * 3 = 2520. The pattern of decreasing multipliers 7, 6, 5, 4, 3 is consistent and natural, confirming the correctness of 2520 as the next term. This structure is reminiscent of partial factorial patterns often used in exam questions.

Why Other Options Are Wrong:
Option 1240 does not equal 840 multiplied by any integer in the descending sequence and breaks the smooth product pattern.Option 2640 corresponds to 840 * 3.1428..., which is not a simple integer multiplier and does not fit the pattern of descending whole number factors.Option 1680 equals 840 * 2, which would continue with a multiplier of 2 instead of the next logical multiplier 3 in the descending sequence 7, 6, 5, 4, 3.

Common Pitfalls:
Some learners initially try to use addition or subtraction, which becomes impractical as the numbers grow rapidly. Others may miscalculate the ratios or overlook the descending nature of the multipliers. Always compute the ratios carefully and look for simple integer sequences such as 7, 6, 5, 4 that describe how the series evolves.

Final Answer:
The missing term that completes the multiplicative series is 2520.