Assertion (A) : If the `n^(th)` term of an A.P. is 7- 4n, then its common differences is -4.
Reason (R ) : Common differences of an A.P .is given by `d=a_(n+1)-a_(n)`

Options

A. Both A and R are true and R is the correct explanation for A.
B. Both A and R are true and R is not correct explanation for A.
C. A is true but R is false.
D. A is false but R is true.

Answer

To solve the problem step by step, we will analyze the assertion and the reason provided in the question.

### Step 1: Identify the nth term of the A.P.
The nth term of the A.P. is given as:
\[ a_n = 7 - 4n \]

### Step 2: Find the (n+1)th term of the A.P.
To find the (n+1)th term, we replace \( n \) with \( n + 1 \):
\[ a_{n+1} = 7 - 4(n + 1) \]
\[ a_{n+1} = 7 - 4n - 4 \]
\[ a_{n+1} = 3 - 4n \]

### Step 3: Calculate the common difference (d).
The common difference \( d \) of an A.P. is given by:
\[ d = a_{n+1} - a_n \]
Substituting the values we found:
\[ d = (3 - 4n) - (7 - 4n) \]
\[ d = 3 - 4n - 7 + 4n \]
\[ d = 3 - 7 \]
\[ d = -4 \]

### Step 4: Conclusion about the assertion.
The assertion states that the common difference is -4, which we have calculated to be true.

### Step 5: Analyze the reason.
The reason states that the common difference of an A.P. is given by \( d = a_{n+1} - a_n \), which is indeed correct.

### Final Conclusion:
Both the assertion and the reason are true, and the reason correctly explains the assertion.