Question

Find the next number in the series: 3, 8, 15, 24, 35, ?

• A. 46
• B. 48
• C. 50
• D. 52

Hint:

Whenever a number series has terms that are close to perfect squares (\(4, 9, 16, 25, 36, 49, \dots\)), always check for the pattern \((n^2 - 1)\) or \((n^2 + 1)\). It saves valuable time during the exam.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This is a number series problem where we are given a sequence of numbers (3, 8, 15, 24, 35). We need to analyze the progression of terms to determine the underlying rule and compute the next term in the sequence.

Step 2: Key Formula or Approach:
• We can test two primary pattern-seeking approaches:
- Approach 1: Check the differences between successive terms to identify any arithmetic pattern.
- Approach 2: Check if the terms can be represented as an algebraic function, such as perfect squares minus a constant.

Step 3: Detailed Explanation:

• Method 1: Analysis of Differences
Let us calculate the differences between consecutive terms: - Difference between 1st and 2nd term: \[ 8 - 3 = 5 \] - Difference between 2nd and 3rd term: \[ 15 - 8 = 7 \] - Difference between 3rd and 4th term: \[ 24 - 15 = 9 \] - Difference between 4th and 5th term: \[ 35 - 24 = 11 \] - The differences are 5, 7, 9, 11. These differences form an arithmetic progression of consecutive odd numbers starting from 5.
- The next odd number after 11 is 13.
- Therefore, the next term in the series is: \[ 35 + 13 = 48 \]
• Method 2: Square Pattern Analysis
Let us express each term of the series relative to perfect square numbers: - 1st term: \[ 2^2 - 1 = 4 - 1 = 3 \] - 2nd term: \[ 3^2 - 1 = 9 - 1 = 8 \] - 3rd term: \[ 4^2 - 1 = 16 - 1 = 15 \] - 4th term: \[ 5^2 - 1 = 25 - 1 = 24 \] - 5th term: \[ 6^2 - 1 = 36 - 1 = 35 \] - The general \(n\)-th term of the series can be defined as: \[ T_n = (n + 1)^2 - 1 \quad \text{for } n = 1, 2, 3, \dots \] - To find the 6th term (\(n = 6\)): \[ T_6 = (6 + 1)^2 - 1 = 7^2 - 1 = 49 - 1 = 48 \] Both methods consistently confirm that the next number in the series is 48.

Step 4: Final Answer:
The next number in the series is 48. Hence, the correct option is (B).