Identifying the Series Pattern

The given number series is: $0, 6, 25, 62, $ _________.

We need to find the pattern that generates these numbers based on their position ($n$). Let's denote the terms as $T_n$, where $n=1, 2, 3, 4, ...$

Analyzing Term Relationships

Let's compare each term with the cube of its position ($n^3$):

• For $n=1$: The term is $0$. $1^3 = 1$. The difference is $0 - 1 = -1$. So, $T_1 = 1^3 - 1$.
• For $n=2$: The term is $6$. $2^3 = 8$. The difference is $6 - 8 = -2$. So, $T_2 = 2^3 - 2$.
• For $n=3$: The term is $25$. $3^3 = 27$. The difference is $25 - 27 = -2$. So, $T_3 = 3^3 - 2$.
• For $n=4$: The term is $62$. $4^3 = 64$. The difference is $62 - 64 = -2$. So, $T_4 = 4^3 - 2$.

Deducing the Rule

From the analysis, we observe a pattern:

• The first term ($n=1$) follows the rule $T_1 = n^3 - 1$.
• Subsequent terms ($n \geq 2$) follow the rule $T_n = n^3 - 2$.

This specific pattern deviation for the first term is common in sequence problems.

Calculating the Next Term

To find the next term in the series, we need to calculate $T_5$. Since $n=5$ is greater than or equal to 2, we use the rule $T_n = n^3 - 2$.

For $n=5$: $T_5 = 5^3 - 2$ $T_5 = 125 - 2$ $T_5 = 123$

Thus, the next number in the series is $123$.