This question is from the OG, and I believe it should be b-a and 2^n in the prompt, not b-1 and 2^2. The question should read:
if n is a positive integer, is the value of b-a at least twice the value of 3^n-2^n?
1.a=2^n+1 and b=3^n+1
2.n=3
The prompt can be re-written as:
b-a >= 2*(3^n - 2^n) ?
b-a >= 2*(3^n) - 2^(n+1) ?
Statement 1:
a=2^n+1 and b=3^n+1
Therefore, using the inequality from the prompt, we want to know if it's true that:
3^(n+1) - 2^(n+1) >= 2*(3^n) - 2^(n+1)
Add 2^(n+1) to both sides:
3^(n+1) >= 2*(3^n)
3*(3^n) >= 2*(3^n)
Since we know 3^n is positive, we know we can divide both sides of the inequality by 3^n without the sign changing.
3 >= 2.
THat is always true, and therefore, St (1) is sufficient.
Statement 2:
Nothing about a or b ... insufficient
Rich Zwelling
GMAT Instructor, Veritas Prep
