Question

a, b and c are three positive integers. What is the value of \(a^2+b^2+c^2\)?

Statement 1: \(a^2+b^2=17\) and c is the arithmetic mean of a and b
Statement 2: The geometric mean of a and b is 2

• A. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

Combine \(a^2+b^2\) with \(ab\) to get \((a+b)^2\), then use c as the arithmetic mean.

Answer 1

The Correct Option is C

Step 1: Note what value we need.
We need the single number \( a^2+b^2+c^2 \), where c is defined through a and b once we know it is their arithmetic mean.

Step 2: Test statement 1 alone.
Statement 1 gives \( a^2+b^2=17 \) and says c is the arithmetic mean of a and b, so \( c = \frac{a+b}{2} \).
Knowing only the sum of squares does not fix the sum \( a+b \) by itself; different pairs of numbers can share the same sum of squares while having different sums.
Without a second equation linking a and b, c cannot be pinned to one value. Statement 1 alone is not sufficient.

Step 3: Test statement 2 alone.
Statement 2 gives the geometric mean of a and b as 2, so \( \sqrt{ab} = 2 \), meaning \( ab = 4 \).
This says nothing about \( a^2+b^2 \) and nothing about c at all, since the rule for c comes only from statement 1.
Statement 2 alone is not sufficient.

Step 4: Combine both statements.
From statement 1, \( a^2+b^2=17 \). From statement 2, \( ab=4 \).
Add twice the product to the sum of squares: \( (a+b)^2 = a^2+b^2+2ab = 17+8=25 \), so \( a+b=5 \) (taking the positive root since a and b are positive).
Since a and b are positive integers with \( a^2+b^2=17 \) and \( ab=4 \), the pair works out to \( a=4, b=1 \) (or the reverse), which fits both equations.
Now c, the arithmetic mean of a and b, is \( c = \frac{5}{2} \).
So \( a^2+b^2+c^2 = 17 + \left(\frac{5}{2}\right)^2 = 17 + \frac{25}{4} = \frac{93}{4} \).
Both statements together fix a unique value.

Final Answer:
Neither statement alone gives a fixed value, but together they do. \[ \boxed{\text{Option (c): Both statements together are needed}} \]