Select the option that will replace the question mark (?) in the following number series.
456, 444, 420, 384, 336, ?

Question

Select the option that will replace the question mark (?) in the following number series.456, 444, 420, 384, 336, ?

Prepp Team · Accepted Answer

Analyzing the Number Series Pattern

The given number series is 456, 444, 420, 384, 336, ?. We need to find the next term in this series by identifying the underlying pattern.

Let's look at the differences between consecutive terms:

Difference between the 2nd and 1st term: $444 - 456 = -12$
Difference between the 3rd and 2nd term: $420 - 444 = -24$
Difference between the 4th and 3rd term: $384 - 420 = -36$
Difference between the 5th and 4th term: $336 - 384 = -48$

The sequence of differences is -12, -24, -36, -48.

Now, let's examine the differences between these consecutive differences (the second level of differences):

Difference between the 2nd and 1st difference: $-24 - (-12) = -24 + 12 = -12$
Difference between the 3rd and 2nd difference: $-36 - (-24) = -36 + 24 = -12$
Difference between the 4th and 3rd difference: $-48 - (-36) = -48 + 36 = -12$

We can observe that the second differences are constant and equal to -12. This indicates that the first differences form an arithmetic progression where each term decreases by 12.

Following this pattern, the next difference in the sequence -12, -24, -36, -48 should be $-48 - 12 = -60$.

To find the next term in the original series, we subtract this next difference (-60) from the last term (336):

Next term $= 336 + (\text{next difference})$

Next term $= 336 + (-60)$

Next term $= 336 - 60$

Next term $= 276$

So, the next term in the series is 276.

Term	Value	Difference from Previous Term	Second Difference
1st	456	-	-
2nd	444	$444 - 456 = -12$	-
3rd	420	$420 - 444 = -24$	$-24 - (-12) = -12$
4th	384	$384 - 420 = -36$	$-36 - (-24) = -12$
5th	336	$336 - 384 = -48$	$-48 - (-36) = -12$
6th	?	$-48 + (-12) = -60$	$-12$

The pattern shows that the differences are decreasing by 12 each time. The next difference is -60, so the next term is $336 - 60 = 276$.

Revision Table: Key Concepts for Number Series

Concept	Description
Number Series	A sequence of numbers following a specific pattern or rule.
Difference Method	Finding the difference between consecutive terms to identify an arithmetic or other sequential pattern.
Second Difference	The difference between consecutive differences, useful for identifying quadratic or other higher-order patterns.
Arithmetic Progression (AP)	A sequence where the difference between consecutive terms is constant. The differences in this problem form an AP.

Additional Information on Number Series Patterns

Number series questions test your logical reasoning and numerical ability. Patterns can be based on various operations like addition, subtraction, multiplication, division, squares, cubes, or combinations of these. Sometimes, the pattern is in the differences between terms, or even in the second or third differences.

Common patterns include:

Arithmetic Series: Constant difference.
Geometric Series: Constant ratio.
Square or Cube Series: Terms are squares or cubes of natural numbers, or related to them.
Mixed Series: A combination of two or more patterns.
Alternating Series: Two different patterns followed alternately by the terms.
Fibonacci or similar Series: Each term is the sum of the two preceding terms.

Analyzing the differences between terms is a fundamental technique to solve number series problems, especially when the series is not immediately recognizable as arithmetic or geometric.