inertia2010 wrote:Can someone please explain me this?

x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity 7^x-1 - 5^x?

z < 25 and w = 7^x
x = 4

Thanks!!

x^(odd integer) > 0 implies x is a positive integer. Example: 2 ^ 3 = 8 and (-2)^3 = -8 and since it is given that x^(odd integer) > 0, so x is a positive integer.

w - z > 5(7^x-1 - 5^x) or w - z > 5 * 7^(x - 1) - 5^(x + 1)

(1) z < 25 and w = 7^x
So, the question is 7^x - (a number less than 25) > 5 * 7^(x - 1) - 5^(x + 1)?
It can be seen here that 5^(x + 1) will always be greater than 25, as minimum value of x can be 1, which means minimum value of 5^(x + 1) will be 25 or greater than 25.
So, z < 5^(x + 1) always.

Now, for x = 1, 7^x = 7 and 5 * 7^(1 - 1) = 5
for x = 2, 7^x = 49 and 5 * 7^(2 - 1) = 35
So, it can be seen that 7^x > 5 * 7^(x - 1) always.

Hence, it can be seen that 7^x - (a number less than 25) > 5 * 7^(x - 1) - 5^(x + 1) always holds true, as 7^x > 5 * 7^(x - 1) and z < 5^(x + 1).

Therefore, statement 1 is SUFFICIENT.

(2) x = 4 implies w - z > 5 * 7^(4 - 1) - 5^(4 + 1) or w - z > 5 * 7^3 - 5^5 or w - z > 1715 - 3125
or w - z > -1410
If w = 49 and z = 20, then w - z > -1410 holds true.
If w = -2401 and z = 20, then w - z = -2421 < -1410; here w - z < -1410.

Since we don't get a definite answer, so statement 2 is NOT sufficient.

The correct answer is A.