Which of the following numbers will replace the question mark (?) in the given series?
7, 7, 5, 15, 11, 55, 49, 343, ?

Question

Which of the following numbers will replace the question mark (?) in the given series?7, 7, 5, 15, 11, 55, 49, 343, ?

Prepp Team · Accepted Answer

Analyzing the Number Series Question

The question asks us to find the number that replaces the question mark (?) in the given number series: 7, 7, 5, 15, 11, 55, 49, 343, ?.

Solving number series problems involves identifying the underlying pattern or rule that governs the sequence of numbers. This pattern can involve arithmetic operations (addition, subtraction, multiplication, division), or a combination of these, applied sequentially or in a repeating cycle.

Identifying the Pattern in the Series

Let's examine the relationship between consecutive terms or alternate terms in the series: 7, 7, 5, 15, 11, 55, 49, 343, ?

Observing the numbers, we can see that they don't follow a simple arithmetic or geometric progression. Let's look at the operations applied between consecutive numbers:

From 7 to 7: This could be +0 or ×1.
From 7 to 5: This is -2.
From 5 to 15: This is +10 or ×3.
From 15 to 11: This is -4.
From 11 to 55: This is +44 or ×5.
From 55 to 49: This is -6.
From 49 to 343: This is +294 or ×7.

It appears there are alternating operations. Let's group the operations based on the type of change:

First to second term: $7 \to 7$ ($\times 1$)
Second to third term: $7 \to 5$ ($- 2$)
Third to fourth term: $5 \to 15$ ($\times 3$)
Fourth to fifth term: $15 \to 11$ ($- 4$)
Fifth to sixth term: $11 \to 55$ ($\times 5$)
Sixth to seventh term: $55 \to 49$ ($- 6$)
Seventh to eighth term: $49 \to 343$ ($\times 7$)
Eighth to ninth term (the missing number): $343 \to ?$ (Apply the next operation)

The operations seem to alternate between multiplication and subtraction. Let's look at the values used in these operations:

Multiplication factors: 1, 3, 5, 7... This is a sequence of consecutive odd numbers. The next multiplication factor would be 9.
Subtraction values: 2, 4, 6... This is a sequence of consecutive even numbers. The next subtraction value would be 8.

The sequence of operations is: $\times 1, - 2, \times 3, - 4, \times 5, - 6, \times 7, - 8, \times 9, \dots$

The last operation applied was multiplication by 7 ($49 \times 7 = 343$). The next operation in the series should be subtraction of 8.

Applying the Pattern to Find the Missing Term

The last known number in the series is 343. According to the identified pattern, the next step is to subtract 8 from this number to find the missing term (?).

Missing term $= 343 - 8$

$$343 - 8 = 335$$

So, the number that replaces the question mark (?) is 335.

Verification of the Pattern

Let's trace the series with the identified pattern:

Start with 7.
$7 \times 1 = 7$ (Second term)
$7 - 2 = 5$ (Third term)
$5 \times 3 = 15$ (Fourth term)
$15 - 4 = 11$ (Fifth term)
$11 \times 5 = 55$ (Sixth term)
$55 - 6 = 49$ (Seventh term)
$49 \times 7 = 343$ (Eighth term)
$343 - 8 = 335$ (Ninth term, the missing number)

The pattern fits the given series perfectly, leading to 335 as the next term.

Revision Table: Key Concepts in Number Series

Pattern Type	Description	Example
Arithmetic Progression (AP)	Constant difference between terms.	2, 5, 8, 11... (difference = 3)
Geometric Progression (GP)	Constant ratio between terms.	3, 6, 12, 24... (ratio = 2)
Difference Series	Differences between consecutive terms form a pattern (AP, GP, etc.).	1, 2, 4, 7, 11... (differences: 1, 2, 3, 4)
Alternating Series	Pattern alternates between two different rules or sequences.	Identified pattern in this question ($ \times $ odd, $ - $ even).
Square/Cube Series	Terms related to squares or cubes of numbers.	1, 4, 9, 16... ($1^2, 2^2, 3^2, 4^2$)
Combined Operations	Each step involves a combination of operations (e.g., $\times 2 + 1$).	2, 5, 11, 23... ($2 \times 2 + 1 = 5$, $5 \times 2 + 1 = 11$)

Additional Information on Series Completion

Number series questions are common in logical reasoning and quantitative aptitude sections of various exams. To solve them effectively, consider the following strategies:

Calculate differences between consecutive terms. If not constant, calculate differences between differences (second-order differences).
Look for ratios between consecutive terms.
Check for alternating patterns, where the rule changes for every alternate term or group of terms.
Consider patterns involving squares, cubes, prime numbers, or Fibonacci sequence.
Try combinations of operations, like multiplication followed by addition or subtraction.
Break down complex series into two simpler interleaved series.
Practice with various types of series to improve recognition of common patterns.

Recognizing the type of pattern, such as the alternating multiplication and subtraction pattern seen in this question, is key to finding the missing number quickly and accurately.