Choose the correct choice in the following and justify :
`30^(th)` term of the A.P. : 10, 7, 4, ….., is

Options

A. 97
B. 44
C. `-77`
D. `-87`

Answer

To find the 30th term of the arithmetic progression (A.P.) given by the sequence 10, 7, 4, ..., we can follow these steps:

### Step 1: Identify the first term (A) and the common difference (D)
- The first term \( A \) is the first number in the sequence, which is \( 10 \).
- To find the common difference \( D \), we subtract the first term from the second term:
\[ D = 7 - 10 = -3 \]

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term of an A.P. is given by:
\[ A_n = A + (n-1)D \]
where \( A_n \) is the nth term, \( A \) is the first term, \( n \) is the term number, and \( D \) is the common difference.

### Step 3: Substitute the values into the formula
We need to find the 30th term, so we set \( n = 30 \):
\[ A_{30} = A + (30-1)D \]
Substituting the values of \( A \) and \( D \):
\[ A_{30} = 10 + (29)(-3) \]

### Step 4: Calculate the value
Now, we perform the multiplication and addition:
\[ A_{30} = 10 + (29 \times -3) = 10 - 87 = -77 \]

### Conclusion
Thus, the 30th term of the A.P. is:
\[ \boxed{-77} \]