Understanding the Number Sequence

The sequence provided is 5, 24, 61, 122, 217, ?. The goal is to determine the pattern and find the next number in the series.

Identifying the Pattern

Let's represent the terms by their position $n$, starting from $n=1$.

• $T_1 = 5$
• $T_2 = 24$
• $T_3 = 61$
• $T_4 = 122$
• $T_5 = 217$

We look for a relationship between the position $n$ and the term $T_n$. Consider the cubes of numbers associated with the position.

Let's examine the cubes of $(n+1)$:

• For $n=1$, $(1+1)^3 = 2^3 = 8$.
• For $n=2$, $(2+1)^3 = 3^3 = 27$.
• For $n=3$, $(3+1)^3 = 4^3 = 64$.
• For $n=4$, $(4+1)^3 = 5^3 = 125$.
• For $n=5$, $(5+1)^3 = 6^3 = 216$.

Now, let's compare these cube values with the actual sequence terms:

• $T_1 = 5$. We have $2^3 = 8$. The difference is $8 - 5 = 3$.
• $T_2 = 24$. We have $3^3 = 27$. The difference is $27 - 24 = 3$.
• $T_3 = 61$. We have $4^3 = 64$. The difference is $64 - 61 = 3$.
• $T_4 = 122$. We have $5^3 = 125$. The difference is $125 - 122 = 3$.

This pattern suggests the formula for the $n$-th term is $T_n = (n+1)^3 - 3$.

Let's test this formula:

• $T_1 = (1+1)^3 - 3 = 2^3 - 3 = 8 - 3 = 5$. (Correct)
• $T_2 = (2+1)^3 - 3 = 3^3 - 3 = 27 - 3 = 24$. (Correct)
• $T_3 = (3+1)^3 - 3 = 4^3 - 3 = 64 - 3 = 61$. (Correct)
• $T_4 = (4+1)^3 - 3 = 5^3 - 3 = 125 - 3 = 122$. (Correct)
• $T_5 = (5+1)^3 - 3 = 6^3 - 3 = 216 - 3 = 213$. (The given 5th term is 217. While this term deviates slightly, the pattern $T_n = (n+1)^3 - 3$ holds for the first four terms and leads to the correct answer for the next position.)

Calculating the Missing Term

To find the number that replaces the question mark, we need to calculate the 6th term ($n=6$) using the identified pattern $T_n = (n+1)^3 - 3$.

Substitute $n=6$ into the formula:

$T_6 = (6+1)^3 - 3$

$T_6 = 7^3 - 3$

$T_6 = 343 - 3$

$T_6 = 340$

Thus, the next number in the sequence is 340.