Series Pattern Analysis

We are asked to find the missing number in the given number series: 14, 33, 71, 147, 299, ?.

To solve this, let's analyze the relationship between consecutive terms in the series.

Identifying the Pattern

Let's examine the difference or a multiplicative relationship between the terms:

• From 14 to 33: The difference is $33 - 14 = 19$.
• From 33 to 71: The difference is $71 - 33 = 38$.
• From 71 to 147: The difference is $147 - 71 = 76$.
• From 147 to 299: The difference is $299 - 147 = 152$.

Observing the differences (19, 38, 76, 152), we can see a pattern: each difference is double the previous one ($19 \times 2 = 38$, $38 \times 2 = 76$, $76 \times 2 = 152$).

Following this pattern, the next difference should be $152 \times 2 = 304$.

Calculating the Missing Number

To find the missing number (the term after 299), we add the calculated difference (304) to the last known term (299):

Missing Number = $299 + 304 = 603$.

Alternative Pattern Verification

Another way to look at the pattern is using a recursive formula. Let $a_n$ be the $n$-th term in the series. We can test a relationship like $a_{n+1} = k \cdot a_n + c$.

• $a_1 = 14$
• $a_2 = 33$. Let's test $a_2 = 2 \cdot a_1 + 5$. $2 \cdot 14 + 5 = 28 + 5 = 33$. This works.
• $a_3 = 71$. Let's test $a_3 = 2 \cdot a_2 + 5$. $2 \cdot 33 + 5 = 66 + 5 = 71$. This works.
• $a_4 = 147$. Let's test $a_4 = 2 \cdot a_3 + 5$. $2 \cdot 71 + 5 = 142 + 5 = 147$. This works.
• $a_5 = 299$. Let's test $a_5 = 2 \cdot a_4 + 5$. $2 \cdot 147 + 5 = 294 + 5 = 299$. This works.

The identified pattern is $a_{n+1} = 2 \cdot a_n + 5$. This confirms the recurrence relation.

Determining the Next Term

Using the confirmed pattern, the next term ($a_6$) is calculated as:

$a_6 = 2 \cdot a_5 + 5$

$a_6 = 2 \cdot 299 + 5$

$a_6 = 598 + 5$

$a_6 = 603$.

Both methods lead to the same result, confirming that the number 603 correctly replaces the question mark in the series.