Identifying the Number Series Pattern

The question presents a sequence of numbers: 1, 4, 13, 40, 121, ?. We need to identify the pattern governing this series to find the next number.

Let's examine the relationship between consecutive terms:

• From 1 to 4: The difference is $4 - 1 = 3$.
• From 4 to 13: The difference is $13 - 4 = 9$.
• From 13 to 40: The difference is $40 - 13 = 27$.
• From 40 to 121: The difference is $121 - 40 = 81$.

The differences are 3, 9, 27, 81. This sequence of differences represents powers of 3: $3^1, 3^2, 3^3, 3^4$.

Alternatively, let's check if there's a multiplication pattern with an addition/subtraction component:

• $1 \times 3 + 1 = 3 + 1 = 4$
• $4 \times 3 + 1 = 12 + 1 = 13$
• $13 \times 3 + 1 = 39 + 1 = 40$
• $40 \times 3 + 1 = 120 + 1 = 121$

The pattern observed is that each number in the series is obtained by multiplying the previous number by 3 and then adding 1. The rule can be represented as: $N_{k+1} = (N_k \times 3) + 1$, where $N_k$ is the k-th term in the series.

Calculating the Next Number

To find the number that follows 121, we apply the identified pattern:

Next Number $= (\text{Previous Number} \times 3) + 1$

Next Number $= (121 \times 3) + 1$

First, calculate the product:

$121 \times 3 = 363$

Now, add 1:

$363 + 1 = 364$

Therefore, the next number in the series is 364.

Matching with Options

The calculated next number is 364. Comparing this with the given options:

• Option 1: 365
• Option 2: 362
• Option 3: 370
• Option 4: 364

The calculated value matches Option 4.