The image presents two number sequences, labeled (i) and (ii). We need to identify the pattern in each sequence and find the next numbers. Sequence (i): 7, 14, 21, 28, ... Sequence (ii): 1, 3, 7, 15, 31, ...

1. Problem Description

The image presents two number sequences, labeled (i) and (ii). We need to identify the pattern in each sequence and find the next numbers.

Sequence (i): 7, 14, 21, 28, ...

Sequence (ii): 1, 3, 7, 15, 31, ...

2. Solution Steps

Sequence (i): This is an arithmetic progression.

The difference between consecutive terms is:

$14 - 7 = 7$

$21 - 14 = 7$

$28 - 21 = 7$

So the common difference is

7. The next term is obtained by adding 7 to the last term.

$28 + 7 = 35$

Sequence (ii):

Let's analyze the differences between consecutive terms:

$3 - 1 = 2$

$7 - 3 = 4$

$15 - 7 = 8$

$31 - 15 = 16$

The differences are 2, 4, 8, 16, which are powers of 2 (2^1, 2^2, 2^3, 2^4). The next difference should be $2^5 = 32$.

So the next term in the sequence is:

$31 + 32 = 63$

3. Final Answer

Sequence (i): The next number is

3

5. Sequence (ii): The next number is 63.