Statement -1: If `a_(1),a_(2),a_(3), . . . . .,a_(n), . . .` is an A.P. such that `a_(1)+a_(4)+a_(7)+ . . . .+a_(16)=147`, then `a_(1)+a_(6)+a_(11)+a_(16)=98`
Statement -2: In an A.P., the sum of the terms equidistant from the beginning and the end is always same and is equal to the sum of first and last term.

Options

A. Statement -1 is true, Statement -2 is True, Statement -2 is a correct explanation for Statement for Statement -1.
B. Statement -1 is true, Statement -2 is True, Statement -2 is not a correct explanation for Statement for Statement -1.
C. Statement -1 is true, Statement -2 is False.
D. Statement -1 is False, Statement -2 is True.

Explanation

To solve the problem, we need to analyze both statements regarding the arithmetic progression (A.P.) and verify their correctness.

### Step 1: Understanding the A.P. and the given sums
Let the first term of the A.P. be \( a_1 \) and the common difference be \( d \). The terms of the A.P. can be expressed as:
- \( a_1 = a_1 \)
- \( a_2 = a_1 + d \)
- \( a_3 = a_1 + 2d \)
- \( a_4 = a_1 + 3d \)
- \( a_5 = a_1 + 4d \)
- \( a_6 = a_1 + 5d \)
- \( a_7 = a_1 + 6d \)
- \( a_8 = a_1 + 7d \)
- \( a_9 = a_1 + 8d \)
- \( a_{10} = a_1 + 9d \)
- \( a_{11} = a_1 + 10d \)
- \( a_{12} = a_1 + 11d \)
- \( a_{13} = a_1 + 12d \)
- \( a_{14} = a_1 + 13d \)
- \( a_{15} = a_1 + 14d \)
- \( a_{16} = a_1 + 15d \)

### Step 2: Analyzing Statement 1
The first statement gives us the sum:
\[ a_1 + a_4 + a_7 + a_{10} + a_{13} + a_{16} = 147 \]
Substituting the terms:
\[ a_1 + (a_1 + 3d) + (a_1 + 6d) + (a_1 + 9d) + (a_1 + 12d) + (a_1 + 15d) = 147 \]
This simplifies to:
\[ 6a_1 + (3d + 6d + 9d + 12d + 15d) = 147 \]
Calculating the coefficients of \( d \):
\[ 3 + 6 + 9 + 12 + 15 = 45 \]
Thus, we have:
\[ 6a_1 + 45d = 147 \]
Dividing the entire equation by 3:
\[ 2a_1 + 15d = 49 \quad \text{(Equation 1)} \]

### Step 3: Analyzing the second part of Statement 1
Now we need to verify:
\[ a_1 + a_6 + a_{11} + a_{16} = 98 \]
Substituting the terms:
\[ a_1 + (a_1 + 5d) + (a_1 + 10d) + (a_1 + 15d) = 98 \]
This simplifies to:
\[ 4a_1 + (5d + 10d + 15d) = 98 \]
Calculating the coefficients of \( d \):
\[ 5 + 10 + 15 = 30 \]
Thus, we have:
\[ 4a_1 + 30d = 98 \]
Dividing the entire equation by 2:
\[ 2a_1 + 15d = 49 \quad \text{(Equation 2)} \]

### Step 4: Comparing both equations
From Equation 1 and Equation 2, we see that both equations are identical:
\[ 2a_1 + 15d = 49 \]
Thus, the first statement is true.

### Step 5: Verifying Statement 2
Statement 2 states that in an A.P., the sum of terms equidistant from the beginning and the end is always the same and equal to the sum of the first and last terms. This is a well-known property of A.P.s and is indeed true.

### Conclusion
Both statements are true, and Statement 2 correctly explains Statement 1.

### Final Answer
- Statement 1 is true.
- Statement 2 is true.
- Statement 2 is a correct explanation for Statement 1.