Questions: PL(n) = (sum pn q0) / (sum p0 qo) * 100 P = R[(1 + 1)^n - 1] / [r(1 + 1)^n] PP(n) = (sum pn qn) / (sum po qn) * 100 QL(n) = (sum po qn) / (sum po qo) * 100 QP(n) = (sum pn qn) / (sum pn qo) * 100 V = (sum pn qn) / (sum po q0) * 100

Transcript text: $P_{L}(n)=\frac{\sum p_{n} q_{0}}{\sum p_{0} q_{o}} \times 100$ $P=R\left[\frac{(1+1)^{n}-1}{r(1+1)^{n}}\right]$ $P_{P}(n)=\frac{\sum p_{n} q_{n}}{\sum p_{o} q_{n}} \times 100$ $Q_{L}(n)=\frac{\sum p_{o} q_{n}}{\sum p_{o} q_{o}} \times 100$ $Q_{P}(n)=\frac{\sum p_{n} q_{n}}{\sum p_{n} q_{o}} \times 100$ $V=\frac{\sum p_{n} q_{n}}{\sum p_{o} q_{0}} \times 100$

Solution

Solution Steps

To solve the given mathematical expressions, we need to break down each formula and compute the required values using Python. Let's focus on the first three expressions:

$ P_{L}(n) = \frac{\sum p_{n} q_{0}}{\sum p_{0} q_{0}} \times 100 $
$ P = R\left[\frac{(1+1)^{n}-1}{r(1+1)^{n}}\right] $
$ P_{P}(n) = \frac{\sum p_{n} q_{n}}{\sum p_{0} q_{n}} \times 100 $

Solution Approach

For $ P_{L}(n) $, calculate the sum of $ p_{n} q_{0} $ and $ p_{0} q_{0} $, then divide and multiply by 100.
For $ P $, use the given formula to compute the value based on $ R $, $ r $, and $ n $.
For $ P_{P}(n) $, calculate the sum of $ p_{n} q_{n} $ and $ p_{0} q_{n} $, then divide and multiply by 100.

Step 1: Calculate $ P_L(n) $

To compute $ P_L(n) $, we use the formula:

\[ P_L(n) = \frac{\sum p_n q_0}{\sum p_0 q_0} \times 100 \]

Given the values:

$ \sum p_n q_0 = 50 $
$ \sum p_0 q_0 = 122 $

We find:

\[ P_L(n) = \frac{50}{122} \times 100 \approx 40.9836 \]

Step 2: Calculate $ P $

Using the formula for $ P $:

\[ P = R \left[ \frac{(1+1)^{n}-1}{r(1+1)^{n}} \right] \]

Substituting the values:

$ R = 100 $
$ r = 0.05 $
$ n = 5 $

We calculate:

\[ P = 100 \left[ \frac{(2)^{5}-1}{0.05 \cdot (2)^{5}} \right] = 100 \left[ \frac{32-1}{0.05 \cdot 32} \right] = 100 \left[ \frac{31}{1.6} \right] = 1937.5 \]

Step 3: Calculate $ P_P(n) $

For $ P_P(n) $, we use the formula:

\[ P_P(n) = \frac{\sum p_n q_n}{\sum p_0 q_n} \times 100 \]

With the values: