Evaluate $$\int_{3}^{4} (x + 2) \sqrt{x - 3} \, dx$$ using the substitution $u = x - 3$ - HSC - SSCE Mathematics Extension 1 - Question 12 - 2023 - Paper 1

To prove by induction:

1. Base case: For $n = 1$, LHS = $1 \times 2^2 = 2$, RHS = $2 + (1 - 1)2^{1 + 1} = 2$.

Both sides are equal.

2. Inductive step: Assume true for $n = k$:

$LHS = (1 \times 2^2) + \cdots + (k \times 2^k) = 2 + (k - 1)2^{k + 1}$

Now for $n = k + 1$:

$LHS = (1 \times 2^2) + \cdots + (k \times 2^k) + ((k + 1) \times 2^{k + 1})$

Substituting from the assumption:

$= 2 + (k - 1)2^{k + 1} + (k + 1) \times 2^{k + 1}$

Combining like terms leads to: $= 2 + k \times 2^{k + 1}$ This matches the RHS for $n = k + 1$. Therefore, by induction, the statement holds for all integers $n \geq 1$.

Using the binomial probability formula:

$P(X = k) = C_n^k p^k (1 - p)^{n - k}$

For our case:

• $n = 5$ (treadmills)
• $k = 3$ (in use)
• $p = 0.65$ (used time)
• Therefore, we have:

$P(3) = C_5^3 (0.65)^3 (0.35)^{2}$

Using the binomial formulas for both treadmills and rowing machines:

1. Treadmills (exactly 3 in use):

$P(3) = C_5^3 (0.65)^3 (0.35)^2$

2. Rowing machines (0 in use):

$P(0) = C_4^0 (0.4)^0 (0.6)^{4} = (0.6)^{4}$

Combine results:

$P(3, 0) = P(3) \times P(0) = C_5^3 (0.65)^3 (0.35)^{2} \times (0.6)^{4}$

From the given identity:

$2022C_{30} + 2022C_{31} + 2022C_{32} = 2022C_{32} + 2022C_{31} + 2022C_{30} = 2023C_{32}$

By matching coefficients:

Let $p = 2023$ and $q = 32$. Thus a possible solution is:

$p = 2023, \, q = 32$