The `n^(th)` term of the A.P. a, 3a, 5a, …. Is

Options

A. na
B. `(2n - 1)a`
C. `(2n - 1)a`
D. 2na

Answer

To find the \( n^{th} \) term of the arithmetic progression (A.P.) given as \( a, 3a, 5a, \ldots \), we can follow these steps:

### Step 1: Identify the first term and the common difference
The first term \( a_1 \) of the A.P. is given as \( a \).

To find the common difference \( d \), we subtract the first term from the second term:
\[ d = a_2 - a_1 = 3a - a = 2a \]

### Step 2: Use the formula for the \( n^{th} \) term of an A.P.
The formula for the \( n^{th} \) term \( a_n \) of an A.P. is given by:
\[ a_n = a + (n - 1)d \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 3: Substitute the values into the formula
Now, substituting the values of \( a \) and \( d \) into the formula:
\[ a_n = a + (n - 1)(2a) \]

### Step 4: Simplify the expression
Now, let's simplify the expression:
\[ a_n = a + 2a(n - 1) \]
\[ = a + 2an - 2a \]
\[ = 2an - a \]

### Step 5: Final expression for the \( n^{th} \) term
Thus, the \( n^{th} \) term of the A.P. is:
\[ a_n = (2n - 1)a \]

### Conclusion
The \( n^{th} \) term of the A.P. \( a, 3a, 5a, \ldots \) is \( (2n - 1)a \).

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