The problem presents four different sequences of numbers and asks for an analysis of their patterns.

1. Problem Description

The problem presents four different sequences of numbers and asks for an analysis of their patterns.

2. Solution Steps

Let's analyze each sequence individually:

i) 7, 14, 21, 28, 35, 42, 49

This is an arithmetic sequence where each term is obtained by adding 7 to the previous term.

$a_n = 7n$, where $n$ starts from

1.

ii) 1, 9, 25, 49, 81, 121, 169 (I corrected the last value, which was unclear)

This sequence consists of the squares of odd numbers.

$1 = 1^2$, $9 = 3^2$, $25 = 5^2$, $49 = 7^2$, $81 = 9^2$, $121 = 11^2$, $169=13^2$.

The general term is $a_n = (2n-1)^2$, where $n$ starts from

1.

iii) 1, 3, 7, 15, 31, 63, 127

Each term is close to a power of

2. $1 = 2^1 - 1$, $3 = 2^2 - 1$, $7 = 2^3 - 1$, $15 = 2^4 - 1$, $31 = 2^5 - 1$, $63 = 2^6 - 1$, $127 = 2^7 - 1$.

The general term is $a_n = 2^n - 1$, where $n$ starts from

1.

iv) 2, 5, 10, 17, 26, 37, 50 (I corrected the last value, which was unclear)

The differences between consecutive terms are:

$5-2 = 3$, $10-5 = 5$, $17-10 = 7$, $26-17 = 9$, $37-26 = 11$, $50-37 = 13$.

The differences form an arithmetic sequence. We can express this sequence as $a_n = n^2 + 1$, where $n$ starts from

1. $2 = 1^2 + 1$, $5 = 2^2 + 1$, $10 = 3^2 + 1$, $17 = 4^2 + 1$, $26 = 5^2 + 1$, $37 = 6^2 + 1$, $50 = 7^2 + 1$.

3. Final Answer

i) Arithmetic sequence with $a_n = 7n$.

ii) Squares of odd numbers with $a_n = (2n-1)^2$.

iii) Sequence with $a_n = 2^n - 1$.

iv) Sequence with $a_n = n^2 + 1$.