Question

Find the next number in the series: 5, 6, 14, 45, 184, ?

• A. 845
• B. 755
• C. 965
• D. 925

Hint:

When the numbers in a series are growing rapidly, a multiplication-based pattern (like \(\times n + k\)) is more likely than an addition-based pattern.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to identify the pattern in the given numerical series and find the next term.
Step 2: Key Formula or Approach:
Analyze the relationship between consecutive terms. This often involves operations like addition, subtraction, multiplication, division, or a combination of these.
Step 3: Detailed Explanation:
Let's examine the transition from one number to the next in the series: 5, 6, 14, 45, 184, ...

From 5 to 6: \(5 \times 1 + 1 = 6\)
From 6 to 14: \(6 \times 2 + 2 = 12 + 2 = 14\)
From 14 to 45: \(14 \times 3 + 3 = 42 + 3 = 45\)
From 45 to 184: \(45 \times 4 + 4 = 180 + 4 = 184\) The pattern is clear: to get the next term, we multiply the current term by a sequentially increasing integer (1, 2, 3, 4, ...) and then add that same integer.
The pattern can be expressed as: T\(_{n}\) = T\(_{n-1}\) \(\times\) n + n.
To find the next term after 184, we will use n = 5:
\[ \text{Next Term} = 184 \times 5 + 5 \] \[ \text{Next Term} = 920 + 5 \] \[ \text{Next Term} = 925 \] Step 4: Final Answer:
The next number in the series is 925.