Understanding Number Series Patterns

This question asks us to find the next term in the given number series: 45, 46, 42, 51, 35, ?

To solve a number series question, we need to identify the underlying pattern or rule that governs the sequence of numbers. Let's look at the differences between consecutive terms in the series.

Analyzing the Differences

Let's calculate the difference between each term and the previous term:

• Difference between the 2nd and 1st term: $46 - 45 = +1$
• Difference between the 3rd and 2nd term: $42 - 46 = -4$
• Difference between the 4th and 3rd term: $51 - 42 = +9$
• Difference between the 5th and 4th term: $35 - 51 = -16$

The sequence of differences is: +1, -4, +9, -16.

Identifying the Pattern in Differences

Let's examine the sequence of differences: +1, -4, +9, -16.

We can observe that the absolute values of these differences are 1, 4, 9, 16. These are perfect squares:

• $1 = 1^2$
• $4 = 2^2$
• $9 = 3^2$
• $16 = 4^2$

Also, notice the signs of the differences are alternating: positive, negative, positive, negative.

So, the pattern for the differences is consecutive perfect squares with alternating signs, starting with positive.

The pattern of differences is: $+1^2, -2^2, +3^2, -4^2$.

Predicting the Next Difference

Following this alternating squares pattern, the next difference should be the next perfect square ($5^2$) with the next alternating sign (positive).

The next difference will be $+5^2 = +25$.

Calculating the Missing Term

To find the next term in the series (the one replacing the question mark), we add the predicted difference (+25) to the last term in the given series (35).

Missing term = Last term + Next difference

Missing term = $35 + 25 = 60$

Therefore, the next term in the series is 60.

Step-by-Step Solution Summary

1. Observe the given series: 45, 46, 42, 51, 35, ?.
2. Calculate the difference between successive terms: +1, -4, +9, -16.
3. Identify the pattern in the differences: The differences are consecutive perfect squares ($1^2, 2^2, 3^2, 4^2$) with alternating signs (+, -, +, -). This is $+1^2, -2^2, +3^2, -4^2$.
4. Determine the next difference based on the pattern: The next difference should be $+5^2 = +25$.
5. Add the next difference to the last term of the series to find the missing term: $35 + 25 = 60$.

The completed series is 45, 46, 42, 51, 35, 60.

Based on the identified pattern, the next term is 60.

Revision Table: Number Series Patterns

Additional Information: Solving Number Series

Solving number series problems is a common type of question in aptitude and reasoning tests. Here are some tips for approaching them:

• Always look at the differences between consecutive terms first. This is often the easiest way to spot a pattern.
• If the first level of differences doesn't show a clear pattern, look at the differences of the differences (second-level differences).
• Check for multiplication or division patterns, especially if the numbers are increasing or decreasing rapidly.
• Look for squares, cubes, or other powers. The pattern might involve adding or subtracting these values.
• Consider alternating patterns, where the rule changes for every other term or the operation alternates.
• Sometimes, the pattern might involve a combination of operations, like multiply by 2 then add 1.
• Practice is key! The more series you analyze, the better you become at recognizing common patterns quickly.

This specific problem required recognizing the sequence of differences as alternating squares, which is a common type of number series pattern.