Algebra

Step 1: Understanding the Concept:

We need to simplify the given expression using the formula for the sum of a geometric progression (GP) and then analyze its factors.
Step 2: Key Formula or Approach:

The sum of a geometric series is . The term is divisible by . The term contains as a factor if is even.
Step 3: Detailed Explanation:

Let the given expression be . The term in the bracket is a GP with first term , common ratio , and number of terms (powers from 0 to 125). Substitute this back into the expression for : We are looking for such that is a factor of . Recall the identity . Let . Then: From this, it is clear that is a factor of . Therefore, works. We must check if a larger is possible. The factors of the form for arise from the factorization . The term is a factor. Here . Further splitting or would yield factors with lower powers or odd powers not in the form with . Thus, 63 is the greatest such integer.
Step 4: Final Answer:

The greatest positive integer is 63.

Step 1: Understanding the Concept:

The given function involves an inverse sine function whose argument contains a square root of a quadratic expression. To determine the range, we must ensure that the argument of lies within its valid domain , and then evaluate the possible outputs of the inverse sine function.
Step 2: Key Formula or Approach:

1. The domain of requires .
2. A quadratic expression (with ) attains its minimum at .
3. The function is strictly increasing with respect to .
Step 3: Detailed Explanation:

Let .
For the function to exist, the argument of the inverse sine must satisfy Since is a square root quantity, it is always non-negative. Therefore, Squaring the inequality gives Now consider the quadratic expression inside the root: Completing the square: Hence, the minimum value of is , which occurs at .
Thus, the smallest value of is Combining this with the restriction required for , we obtain Therefore, the argument of the inverse sine varies over the interval .
Applying the inverse sine function to this interval, and using the fact that is increasing, we get Using standard trigonometric values:
Step 4: Final Answer:

The range of the function is .