Find the nth term and the sum to n terms of the sequence: (i) 1+5+13+29+61+.......

Answer

To find the nth term and the sum to n terms of the sequence \(1, 5, 13, 29, 61, \ldots\), we will follow these steps:

### Step 1: Identify the pattern in the sequence
We will start by observing the given terms:
- \(T_1 = 1\)
- \(T_2 = 5\)
- \(T_3 = 13\)
- \(T_4 = 29\)
- \(T_5 = 61\)

### Step 2: Calculate the differences between consecutive terms
Let's calculate the first differences:
- \(T_2 - T_1 = 5 - 1 = 4\)
- \(T_3 - T_2 = 13 - 5 = 8\)
- \(T_4 - T_3 = 29 - 13 = 16\)
- \(T_5 - T_4 = 61 - 29 = 32\)

The first differences are: \(4, 8, 16, 32\).

### Step 3: Calculate the second differences
Now, let's calculate the second differences:
- \(8 - 4 = 4\)
- \(16 - 8 = 8\)
- \(32 - 16 = 16\)

The second differences are: \(4, 8, 16\).

### Step 4: Calculate the third differences
Now, let's calculate the third differences:
- \(8 - 4 = 4\)
- \(16 - 8 = 8\)

The third differences are: \(4, 8\).

### Step 5: Calculate the fourth differences
Now, let's calculate the fourth differences:
- \(8 - 4 = 4\)

The fourth difference is constant, which indicates that the nth term can be represented by a polynomial of degree 4.

### Step 6: Formulate the nth term
Based on the pattern observed, we can deduce that the nth term \(T_n\) can be expressed as:
\[ T_n = 2^n + 1 - 3 \]
Thus, the nth term can be simplified to:
\[ T_n = 2^n - 2 \]

### Step 7: Find the sum of the first n terms
To find the sum of the first n terms \(S_n\), we can express it as:
\[ S_n = T_1 + T_2 + T_3 + \ldots + T_n \]
Substituting \(T_n\):
\[ S_n = (2^1 - 2) + (2^2 - 2) + (2^3 - 2) + \ldots + (2^n - 2) \]
This can be rewritten as:
\[ S_n = (2^1 + 2^2 + 2^3 + \ldots + 2^n) - 2n \]

### Step 8: Use the formula for the sum of a geometric series
The sum of the geometric series \(2^1 + 2^2 + \ldots + 2^n\) can be calculated using the formula:
\[ S = a \frac{r^n - 1}{r - 1} \]
where \(a = 2\) and \(r = 2\):
\[ S_n = 2 \frac{2^n - 1}{2 - 1} = 2(2^n - 1) = 2^{n+1} - 2 \]

### Step 9: Combine the results
Thus, the sum of the first n terms is:
\[ S_n = (2^{n+1} - 2) - 2n = 2^{n+1} - 2 - 2n \]

### Final Result
The nth term of the sequence is:
\[ T_n = 2^n - 2 \]
The sum of the first n terms is:
\[ S_n = 2^{n+1} - 2 - 2n \]