Understanding the Number Series Problem

The question asks us to identify the missing number in the given numerical sequence: $2, 5, 17, 71, ?, 2159$. To solve this, we need to find the underlying pattern or rule that governs the progression of the numbers.

Analyzing the Series Pattern

Let's examine the relationship between consecutive terms in the series:

• From 2 to 5
• From 5 to 17
• From 17 to 71
• From 71 to the missing number (?)
• From the missing number (?) to 2159

Let's try to find a mathematical relationship. We can test multiplication and addition/subtraction rules.

Consider the first term, 2. Let's see how we can get the second term, 5:

• $2 \times 2 + 1 = 4 + 1 = 5$

Now, let's use this potential pattern for the next step, using the second term (5) and the multiplier/adder sequence.

Let the terms be $T_1, T_2, T_3, T_4, T_5, T_6$. The pattern seems to be related to the position of the term.

Let's hypothesize a pattern of the form $T_{n+1} = (T_n \times \text{Multiplier}) + \text{Adder}$.

Checking the steps:

1. Step 1: $T_1 = 2$. To get $T_2 = 5$. Maybe $T_2 = T_1 \times (1+1) + 1 = 2 \times 2 + 1 = 5$.
2. Step 2: $T_2 = 5$. To get $T_3 = 17$. Using the potential pattern: $T_3 = T_2 \times (2+1) + 2 = 5 \times 3 + 2 = 15 + 2 = 17$. This matches.
3. Step 3: $T_3 = 17$. To get $T_4 = 71$. Using the potential pattern: $T_4 = T_3 \times (3+1) + 3 = 17 \times 4 + 3 = 68 + 3 = 71$. This also matches.

The pattern identified is $T_{n+1} = T_n \times (n+1) + n$, where '$n$' represents the position index starting from 1.

Calculating the Missing Number

Now, we apply this pattern to find the missing term ($T_5$) and verify the last term ($T_6$).

To find $T_5$ (the missing number):

• Here, $n=4$.
• $T_5 = T_4 \times (4+1) + 4$
• $T_5 = 71 \times 5 + 4$
• $T_5 = 355 + 4$
• $T_5 = 359$

So, the missing number is 359.

Verifying the Next Term

Let's verify if this pattern holds for the last term ($T_6 = 2159$) using the calculated missing term ($T_5 = 359$).

• Here, $n=5$.
• $T_6 = T_5 \times (5+1) + 5$
• $T_6 = 359 \times 6 + 5$
• $T_6 = 2154 + 5$
• $T_6 = 2159$

This matches the last term provided in the series, confirming our identified pattern.

Conclusion

The pattern is $T_{n+1} = T_n \times (n+1) + n$. Applying this pattern, the missing number in the series $2, 5, 17, 71, ?, 2159$ is 359.