Solving the Number Sequence Puzzle

The question asks us to find the next number in the given sequence: 6, 24, 60, 120, ____.

Sequence Analysis: 6, 24, 60, 120

Let's examine the terms provided in the sequence:

• Term 1: 6
• Term 2: 24
• Term 3: 60
• Term 4: 120

We need to determine the relationship between these terms to predict the 5th term.

Identifying the Pattern Rule

Let's try to find a mathematical pattern. We can check the differences between consecutive terms:

• Difference between Term 2 and Term 1: $24 - 6 = 18$
• Difference between Term 3 and Term 2: $60 - 24 = 36$
• Difference between Term 4 and Term 3: $120 - 60 = 60$

The differences (18, 36, 60) are not constant, so it's not a simple arithmetic sequence. Let's look at the differences between these differences (second differences):

• Second difference 1: $36 - 18 = 18$
• Second difference 2: $60 - 36 = 24$

The second differences (18, 24) are also not constant. Let's check the third differences:

• Third difference: $24 - 18 = 6$

Since the third difference is constant, this suggests a cubic pattern. Let's explore another common pattern type for sequences like this: the product of consecutive numbers.

Consider the formula $T_n = n \times (n+1) \times (n+2)$, where '$n$' is the term number.

• For $n=1$: $T_1 = 1 \times (1+1) \times (1+2) = 1 \times 2 \times 3 = 6$
• For $n=2$: $T_2 = 2 \times (2+1) \times (2+2) = 2 \times 3 \times 4 = 24$
• For $n=3$: $T_3 = 3 \times (3+1) \times (3+2) = 3 \times 4 \times 5 = 60$
• For $n=4$: $T_4 = 4 \times (4+1) \times (4+2) = 4 \times 5 \times 6 = 120$

This formula perfectly matches the given sequence terms.

Calculating the Missing Term

Now, we can use the identified pattern $T_n = n \times (n+1) \times (n+2)$ to find the 5th term ($n=5$):

• For $n=5$: $T_5 = 5 \times (5+1) \times (5+2) = 5 \times 6 \times 7$
• Calculation: $5 \times 6 = 30$, and $30 \times 7 = 210$.

So, the missing term in the sequence is 210.