In the series: 5, 16, 51, 158, ?, which number will replace the question mark?

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Points to Remember:

  • The question requires identifying a pattern in a numerical series.
  • The approach is analytical, focusing on identifying the mathematical relationship between consecutive numbers in the sequence.

Introduction:

This question involves identifying the next number in a given numerical series: 5, 16, 51, 158, ?. This type of question tests pattern recognition and mathematical reasoning skills. Solving it requires analyzing the relationships between consecutive terms to determine the underlying rule governing the sequence.

Body:

Identifying the Pattern:

Let’s examine the differences between consecutive terms:

  • 16 – 5 = 11
  • 51 – 16 = 35
  • 158 – 51 = 107

The differences themselves don’t immediately reveal a simple arithmetic progression. Let’s explore another approach. Let’s look at the relationship between each term and its position in the sequence:

  • Term 1: 5 = 3*1 + 2
  • Term 2: 16 = 35 + 1 = 3(3*1 + 2) + 1
  • Term 3: 51 = 316 + 3 = 3(3*5 + 1) + 3
  • Term 4: 158 = 351 + 5 = 3(3*16 + 3) + 5

Notice a pattern emerging. Each term is approximately three times the previous term, plus an increment that increases by 2 each time (2, 1, 3, 5…). More precisely, the nth term can be expressed recursively as:

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  • an = 3an-1 + (2n – 3) where a1 = 5

Determining the Missing Term:

Using this recursive formula, we can calculate the missing term (the 5th term):

  • a5 = 3a4 + (2*5 – 3) = 3 * 158 + 7 = 474 + 7 = 481

Therefore, the number that replaces the question mark is 481.

Conclusion:

The missing number in the series 5, 16, 51, 158, ? is 481. This was determined by analyzing the relationship between consecutive terms and identifying a recursive formula that accurately describes the sequence. The pattern involves multiplying the previous term by 3 and adding an increment that increases by 2 with each subsequent term. This analytical approach highlights the importance of systematically exploring different mathematical relationships to uncover the underlying pattern in numerical sequences. Further exploration of similar sequences could involve investigating other types of recursive or iterative relationships to enhance mathematical reasoning skills.

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