Analyzing the Number Sequence Pattern

The question asks us to find the next number in the sequence: 75, 76, 72, 81, 65, ?

To solve this, we need to identify the pattern or rule that governs the sequence. Let's look at the difference between consecutive terms:

• Difference between the 2nd and 1st term: $76 - 75 = +1$
• Difference between the 3rd and 2nd term: $72 - 76 = -4$
• Difference between the 4th and 3rd term: $81 - 72 = +9$
• Difference between the 5th and 4th term: $65 - 81 = -16$

Let's examine these differences: +1, -4, +9, -16. Do these numbers suggest a pattern?

We can observe that these differences are related to square numbers:

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

The pattern of the differences seems to be the squares of consecutive integers (1, 2, 3, 4, ...), with alternating signs (+, -, +, -). The sequence of differences is $1^2, -2^2, 3^2, -4^2, \dots$

Following this pattern, the next difference should be $+5^2$.

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

To find the next number in the original sequence, we add this difference to the last term (65):

Next number = Last term + Next difference

Next number = $65 + 25$

Next number = $90$

Therefore, the number that replaces the question mark is 90.

Step-by-Step Calculation

Let the sequence be $a_1, a_2, a_3, a_4, a_5, a_6, \dots$

Given sequence: $a_1=75, a_2=76, a_3=72, a_4=81, a_5=65, a_6=?$

Calculate the differences between consecutive terms:

• $a_2 - a_1 = 76 - 75 = +1$
• $a_3 - a_2 = 72 - 76 = -4$
• $a_4 - a_3 = 81 - 72 = +9$
• $a_5 - a_4 = 65 - 81 = -16$

Observe the pattern in the differences:

• Difference 1: $+1 = +1^2$
• Difference 2: $-4 = -2^2$
• Difference 3: $+9 = +3^2$
• Difference 4: $-16 = -4^2$

The pattern for the difference between $a_n$ and $a_{n-1}$ appears to be $(-1)^{n} \times (n-1)^2$ for $n \ge 2$.

For $n=6$, the difference $a_6 - a_5$ should be $(-1)^6 \times (6-1)^2 = +1 \times 5^2 = +25$.

So, $a_6 - a_5 = +25$.

$a_6 = a_5 + 25$

$a_6 = 65 + 25$

$a_6 = 90$

Summary of the Pattern

The sequence is generated by adding or subtracting consecutive square numbers ($1^2, 2^2, 3^2, 4^2, 5^2, \dots$) to the previous term, with the sign alternating starting from positive.

Adding $+25$ to the 5th term (65) gives the 6th term:

$65 + 25 = 90$

Conclusion

The pattern in the sequence is based on adding and subtracting consecutive square numbers. Following this pattern, the number that replaces the question mark is 90.

Revision Table: Common Sequence Patterns

Additional Information: Solving Number Series Questions

Number series questions are common in competitive exams and aptitude tests. They assess logical reasoning and pattern recognition skills. Here are some tips for solving them:

• Look at the differences between consecutive terms. Are they constant (AP)? Do they form another recognizable sequence (AP, GP, squares, cubes, etc.)?
• Look at the ratio between consecutive terms. Is it constant (GP)?
• Check for alternating patterns, either in the values themselves or the operation used (+/-).
• Consider squares, cubes, or prime numbers.
• Sometimes the pattern involves two interlinked sequences.
• If the numbers are large or grow/shrink rapidly, consider multiplication, division, squares, or cubes.
• If the numbers change slowly, consider addition or subtraction.
• Write down the differences or ratios clearly to spot the underlying pattern.