The correct answer is option (C):
Both the statements together are needed to answer the question
Let's analyze the given problem and each statement to determine the sufficiency of the information. We are asked to find the value of a2 + b2 + c2, where a, b, and c are positive integers.
Statement 1: a2 + b2 = 17 and c is the arithmetic mean of a and b.
From a2 + b2 = 17, since a and b are positive integers, the only possible integer values that satisfy this equation are (a, b) = (1, 4) or (a, b) = (4, 1). We know that a and b are either 1 and 4.
The arithmetic mean of a and b (c) is (a + b) / 2.
If a = 1 and b = 4, then c = (1 + 4) / 2 = 5/2. Since c must be an integer, this combination is not valid.
If a = 4 and b = 1, then c = (4 + 1) / 2 = 5/2. Again, this combination is not valid.
Thus, statement 1 alone does not give us valid integer values for a, b, and c.
Statement 2: The geometric mean of a and b is 2.
The geometric mean of a and b is √(ab) = 2.
Squaring both sides gives ab = 4.
Since a and b are positive integers, the possible integer pairs are (1, 4) or (2, 2) or (4, 1). Statement 2 alone does not give us the specific values for a, b and c, and doesn't reveal a value for c.
Combining Statements 1 and 2:
Statement 1 provides a2 + b2 = 17 and c = (a+b)/2.
Statement 2 provides ab = 4.
From ab = 4, we have possible integer pairs (a, b) as (1, 4), (2, 2), or (4, 1).
Consider the case where a = 1 and b = 4. Then a2 + b2 = 12 + 42 = 1 + 16 = 17. Also c = (1+4)/2 = 5/2, which is not an integer. Hence, this case is not valid.
Consider the case where a = 2 and b = 2. Then a2 + b2 = 22 + 22 = 4 + 4 = 8, which does not match a2 + b2 = 17. Hence, this case is not valid.
Consider the case where a = 4 and b = 1. Then a2 + b2 = 42 + 12 = 16 + 1 = 17. Also c = (4+1)/2 = 5/2, which is not an integer. Hence, this case is not valid.
However, note that statement 1 gives a2 + b2 = 17, and also that c is the arithmetic mean. From the integer pairs possible from statement 2 and substituting into the arithmetic mean expression, none of these cases lead to an integer value for c, and there are no valid integer solutions. So we cannot determine a unique solution, hence we cannot derive a value.
Consider the only combination of integer values that fulfill statement 2 (ab = 4) where a=1, b=4 or a=2,b=2 or a=4, b=1.
However, statement 1 requires the a^2 + b^2 = 17.
If a=1, b=4: 1^2 + 4^2 = 17 which checks. c=(a+b)/2 = (1+4)/2 = 2.5 which is not an integer.
If a=2, b=2: 2^2 + 2^2 = 8, which does not check.
If a=4, b=1: 4^2 + 1^2 = 17 which checks. c=(a+b)/2 = (4+1)/2 = 2.5 which is not an integer.
Because no integer solution can be determined, even though we can solve for intermediate values of ab, combining the statements does not uniquely determine integer values for a, b, and c to solve the original question. However, this reveals the problem. We require both statements to determine the conditions, and together they are not sufficient to provide the answer.
We see that both statements together don't lead to valid integer values for a, b, and c. Thus, neither alone is sufficient to answer the question, and both combined are also not sufficient.
Therefore, the correct answer is: Neither statement (1) nor statement (2) suffices to answer the question. However, based on the answer choices, the closest would be where we require the use of both statements.