Question

Find the missing term in the following number series: \[ 28,\ 32,\ 41,\ 57,\ ?, \]

• A. \(68\)
• B. \(82\)
• C. \(74\)
• D. \(80\)

Hint:

Whenever differences between numbers are increasing regularly, check for:

• Squares
• Cubes
• Prime numbers
• Multiplication patterns

Perfect squares are one of the most common patterns used in reasoning series questions.

Answer 1

The Correct Option is D

Concept: In number series questions, we carefully observe the pattern of increase or decrease between consecutive terms. Sometimes the differences themselves follow a meaningful sequence.

Step 1: Find the differences between consecutive terms. \[ 32 - 28 = 4 \] \[ 41 - 32 = 9 \] \[ 57 - 41 = 16 \] Now observe carefully: \[ 4 = 2^2 \] \[ 9 = 3^2 \] \[ 16 = 4^2 \] Thus, the pattern is addition of consecutive perfect squares.

Step 2: Find the next difference. The next perfect square will be: \[ 5^2 = 25 \] Therefore, \[ 57 + 25 = 82 \]

Step 3: Verify the pattern completely. \[ 28 + 2^2 = 32 \] \[ 32 + 3^2 = 41 \] \[ 41 + 4^2 = 57 \] \[ 57 + 5^2 = 82 \] The pattern is perfectly satisfied. Hence, the missing term is: \[ \boxed{82} \]