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

Step 1: Isolate the term with n. The given equation is: 1 - ((1)/(3))^n = (242)/(243) Subtract (242)/(243) from 1: ((1)/(3))^n = 1 - (242)/(243) ((1)/(3))^n = (243)/(243) - (242)/(243) ((1)/(3))^n = (1)/(243) Step 2: Express the right side as a power of (1)/(3). 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, (1)/(243) can be written as (1)/(3^5), which is equivalent to ((1)/(3))^5. Step 3: Equate the exponents. ((1)/(3))^n = ((1)/(3))^5 Since the bases are equal, the exponents must be equal: n = 5 The value of n is 5. Step 1: Factor the numerator. The numerator is (y+1)(4k^2 - ky). Factor out k from the second term (4k^2 - ky): (y+1)k(4k - y) Step 2: Factor the denominator. The denominator is (y^2 + y - 4ky - 4k)e. Group the terms in the parenthesis: (y(y+1) - 4k(y+1))e Factor out (y+1): (y+1)(y - 4k)e Step 3: Substitute the factored expressions back into the fraction. ((y+1)k(4k - y))/((y+1)(y - 4k)e) Step 4: Simplify the expression. Notice that (4k - y) is the negative of (y - 4k), i.e., (4k - y) = -(y - 4k). Substitute this into the expression: ((y+1)k(-(y - 4k)))/((y+1)(y - 4k)e) Assuming y ≠ -1 and y ≠ 4k, we can cancel out the common terms (y+1) and (y - 4k): (-k)/(e) The simplified expression is -(k)/(e). Step 1: Define the two-digit number using its digits. 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. Step 2: Use the first condition to form an equation. "The product of the digits is 20." t × u = 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. Step 3: Use the second condition to form another equation. "When the digits are reversed, the number so formed is greater than the original number by 9." The reversed number is 10u + t. 10u + t = (10t + u) + 9 Simplify the equation: 10u - u = 10t - t + 9 9u = 9t + 9 Divide by 9: u = t + 1 (**) Step 4: Check which of the possible numbers from Step 2 satisfies equation (**). 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 45. That's 2 down. 3 left today — send the next one.