Solve for n: 1 - (1/3)ⁿ = 242/243. Simplify the expression: (y+1)(4k2-ky) / (y2+y-4ky-4k)e.

The given equation is:

Subtract $\frac{242}{243}$ from 1:

We need to find what power of 3 equals 243.
$3^1 = 3$
$3^2 = 9$
$3^3 = 27$
$3^4 = 81$
$3^5 = 243$
So, $\frac{1}{243}$ can be written as $\frac{1}{3^5}$, which is equivalent to $\left(\frac{1}{3}\right)^5$.

Since the bases are equal, the exponents must be equal:

The value of $n$ is $\boxed{5}$.

The numerator is $(y+1)(4k^2 - ky)$.
Factor out $k$ from the second term $(4k^2 - ky)$:

The denominator is $(y^2 + y - 4ky - 4k)e$.
Group the terms in the parenthesis:

Factor out $(y+1)$:

Notice that $(4k - y)$ is the negative of $(y - 4k)$, i.e., $(4k - y) = -(y - 4k)$.
Substitute this into the expression:

Assuming $y \neq -1$ and $y \neq 4k$, we can cancel out the common terms $(y+1)$ and $(y - 4k)$:

The simplified expression is $\boxed{-\frac{k}{e}}$.

Let the two-digit number be $10t + u$, where $t$ is the tens digit and $u$ is the units digit.
$t$ must be an integer from 1 to 9.
$u$ must be an integer from 0 to 9.

"The product of the digits is 20."

Possible integer pairs $(t, u)$ that satisfy this condition are:

• If $t=4$, then $u=5$. The number is 45.
• If $t=5$, then $u=4$. The number is 54.

"When the digits are reversed, the number so formed is greater than the original number by 9."
The reversed number is $10u + t$.

Simplify the equation:

Divide by 9:

Case 1: For the number 45, $t=4$ and $u=5$.
Substitute into $u = t+1$:
$5 = 4 + 1$
$5 = 5$
This is true, so 45 is a valid number.

Case 2: For the number 54, $t=5$ and $u=4$.
Substitute into $u = t+1$:
$4 = 5 + 1$
$4 = 6$
This is false, so 54 is not the number.

The number is $\boxed{45}$.