What is the nth term of the sequence 1,5,9,13,17. . .. ?

Options

A. 2n-1
B. 2n+1
C. 4n-3
D. None of these

Answer

To find the nth term of the sequence 1, 5, 9, 13, 17, we can follow these steps:

### Step 1: Identify the first term and the common difference
The first term (A) of the sequence is 1. To find the common difference (D), we can subtract the first term from the second term:
- D = 5 - 1 = 4

### Step 2: Confirm the common difference
We can confirm that the common difference is consistent throughout the sequence:
- 9 - 5 = 4
- 13 - 9 = 4
- 17 - 13 = 4

Since the difference is the same, we can conclude that this sequence is an arithmetic progression (AP).

### Step 3: Use the formula for the nth term of an arithmetic progression
The formula for the nth term (A_n) of an arithmetic progression is given by:
\[ A_n = A + (n - 1) \cdot D \]
Where:
- A is the first term
- D is the common difference
- n is the term number

### Step 4: Substitute the known values into the formula
Now we can substitute A = 1 and D = 4 into the formula:
\[ A_n = 1 + (n - 1) \cdot 4 \]

### Step 5: Simplify the expression
Now, we simplify the expression:
\[ A_n = 1 + 4(n - 1) \]
\[ A_n = 1 + 4n - 4 \]
\[ A_n = 4n - 3 \]

### Conclusion
Thus, the nth term of the sequence is:
\[ A_n = 4n - 3 \]

---