In an A.P. 10, 7, 4 ,…….., 30th term is :

Options

A. 97
B. 77
C. `-77`
D. None of these

Answer

To find the 30th term of the arithmetic progression (A.P.) given as 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 \( T_n \) of an A.P. is given by:
\[ T_n = a + (n - 1) \cdot d \]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

### Step 3: Substitute the values into the formula
We need to find the 30th term, so we substitute \( n = 30 \), \( a = 10 \), and \( d = -3 \) into the formula:
\[ T_{30} = 10 + (30 - 1) \cdot (-3) \]

### Step 4: Simplify the expression
Calculate \( (30 - 1) \):
\[ T_{30} = 10 + 29 \cdot (-3) \]
Now calculate \( 29 \cdot (-3) \):
\[ T_{30} = 10 - 87 \]
Finally, simplify:
\[ T_{30} = -77 \]

### Conclusion
Thus, the 30th term of the A.P. is \( -77 \).

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