If the nth term of an A.P. is 4n-1 , find the 30th term and the sum of first 30 terms.

Options

A. 119 and 1830
B. 69 and 1665
C. 119 and 1380
D. none of these

Answer

To solve the problem step by step, we will first find the 30th term of the arithmetic progression (A.P.) given by the nth term formula \( a_n = 4n - 1 \). After that, we will calculate the sum of the first 30 terms.

### Step 1: Find the 30th Term
The nth term of the A.P. is given by the formula:
\[ a_n = 4n - 1 \]
To find the 30th term (\( a_{30} \)), we substitute \( n = 30 \) into the formula:
\[ a_{30} = 4(30) - 1 \]
Calculating this gives:
\[ a_{30} = 120 - 1 = 119 \]

### Step 2: Find the First Term
Next, we need to find the first term (\( a_1 \)) of the A.P. using the same formula:
\[ a_1 = 4(1) - 1 \]
Calculating this gives:
\[ a_1 = 4 - 1 = 3 \]

### Step 3: Calculate the Sum of the First 30 Terms
The formula for the sum of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (a + l) \]
where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term (which in this case is \( a_{30} \)).

Here, we have:
- \( n = 30 \)
- \( a = a_1 = 3 \)
- \( l = a_{30} = 119 \)

Substituting these values into the formula gives:
\[ S_{30} = \frac{30}{2} \times (3 + 119) \]
Calculating this step-by-step:
1. Calculate \( 3 + 119 = 122 \)
2. Calculate \( \frac{30}{2} = 15 \)
3. Now multiply: \( S_{30} = 15 \times 122 \)

Calculating \( 15 \times 122 \):
\[ S_{30} = 1830 \]

### Final Results
- The 30th term is \( 119 \).
- The sum of the first 30 terms is \( 1830 \).