Advanced MathVery hardSimplifying a quotient of shifted powers
Advanced Math practice question
For every positive integer n, the expression (2^(n + 2) − 2ⁿ)/2^(n + 1) is equal to a constant. What is that constant?
Answer: 3/2 or 1.5
2^(n + 2) − 2ⁿ = 2ⁿ(4 − 1) = 3 · 2ⁿ, and dividing by 2^(n + 1) = 2 · 2ⁿ leaves 3/2.
Common wrong answers
- If you answered 1
- 1 comes from subtracting exponents, treating 2^(n + 2) − 2ⁿ as 2^(n + 1).
- If you answered 2
- 2 treats the numerator as 2^(n + 2) alone.
- If you answered 3
- 3 forgets to divide by the extra factor 2 in the denominator.
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