Question

Select the number from the given options that can replace the question mark (?) in the following series.
2, 5, 10, ?, 26

• A. 19
• B. 17
• C. 15
• D. 16

Hint:

When you see numbers like 2, 5, 10, 17, 26, 37, etc., in a series, immediately check for the pattern \(n^2 + 1\) or \(n^2 - 1\), as it is very common.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to identify the pattern in the given number series to find the missing term.
Step 2: Key Formula or Approach:
There are two common ways to solve this series: 1. Find the difference between consecutive terms and see if the differences follow a pattern. 2. Check if the terms in the series relate to squares or cubes of natural numbers.
Step 3: Detailed Explanation:
Method 1: Difference Pattern
Let's find the difference between consecutive terms:
- \(5 - 2 = 3\)
- \(10 - 5 = 5\)
The differences are 3 and 5. These are consecutive odd numbers. The next difference in this pattern should be 7.
Let's find the missing term using this difference:
- \(10 + 7 = 17\)
Now, let's verify the pattern by checking the next term. The next difference should be 9.
- \(17 + 9 = 26\). This matches the last term of the series.
So, the pattern of differences (3, 5, 7, 9) is consistent.
Method 2: Square Pattern
Let's examine each term to see if it relates to a square number.
- \(2 = 1^2 + 1\)
- \(5 = 2^2 + 1\)
- \(10 = 3^2 + 1\)
The pattern appears to be \(n^2 + 1\), where n starts from 1.
The missing term would be for n=4:
- \(4^2 + 1 = 16 + 1 = 17\)
Let's verify the next term for n=5:
- \(5^2 + 1 = 25 + 1 = 26\). This matches the last term.
Both methods confirm the missing number.
Step 4: Final Answer:
The missing number in the series is 17.