Number Pattern Calculation: Finding the Value for 8

The question presents a sequence where a number '$n$' relates to a result. We need to identify the pattern to find the result for '$n=8$'.

Let's analyze the given pairs:

• For $n=2$, the result is 12.
• For $n=3$, the result is 36.
• For $n=4$, the result is 80.
• For $n=5$, the result is 150.
• For $n=6$, the result is 252.
• For $n=7$, the result is 392.

Identifying the Number Pattern Logic

We observe that the result can be expressed as the input number '$n$' multiplied by a certain factor. Let's find this factor for each case:

• $12 \div 2 = 6$
• $36 \div 3 = 12$
• $80 \div 4 = 20$
• $150 \div 5 = 30$
• $252 \div 6 = 42$
• $392 \div 7 = 56$

The sequence of factors is 6, 12, 20, 30, 42, 56.

Now, let's find the pattern in this sequence of factors. Notice that each factor '$m$' can be represented as '$n \times (n+1)$':

• For $n=2$, factor is $2 \times (2+1) = 2 \times 3 = 6$.
• For $n=3$, factor is $3 \times (3+1) = 3 \times 4 = 12$.
• For $n=4$, factor is $4 \times (4+1) = 4 \times 5 = 20$.
• For $n=5$, factor is $5 \times (5+1) = 5 \times 6 = 30$.
• For $n=6$, factor is $6 \times (6+1) = 6 \times 7 = 42$.
• For $n=7$, factor is $7 \times (7+1) = 7 \times 8 = 56$.

This pattern holds true. Therefore, the overall relationship is: Result $= n \times (\text{factor}) = n \times [n \times (n+1)] = n^2 \times (n+1)$.

Calculating the Result for 8

Using the derived formula, we can calculate the result for $n=8$:

Result $= n^2 \times (n+1)$

Substitute $n=8$:

Result $= 8^2 \times (8+1)$

Result $= 64 \times 9$

Result $= 576$

Thus, for $n=8$, the result is 576.