Math Problem Statement

If p_{t+1} = 0.96p_t and p_0 = 14, what is p_{69}? If p_{t+1} = p_t + 0.13 and p_0 = 20, what is p_{100}?

Solution

Let’s analyze and solve the problem step by step:

Part (a):

The recurrence relation is: pt+1=0.96ptp_{t+1} = 0.96p_t and the initial condition is p0=14p_0 = 14. You need to calculate p69p_{69}.

General solution for the recurrence:

This is a geometric progression. The general form of ptp_t is: pt=p0(0.96)tp_t = p_0 \cdot (0.96)^t

Substituting p0=14p_0 = 14 and t=69t = 69: p69=14(0.96)69p_{69} = 14 \cdot (0.96)^{69}

Computing p69p_{69}:

Let’s calculate: p6914(0.96)69p_{69} \approx 14 \cdot (0.96)^{69}

Part (b):

The recurrence relation is: pt+1=pt+0.13p_{t+1} = p_t + 0.13 and the initial condition is p0=20p_0 = 20. You need to calculate p100p_{100}.

General solution for the recurrence:

This is an arithmetic progression where each term increases by 0.13. The general form is: pt=p0+t0.13p_t = p_0 + t \cdot 0.13

Substituting p0=20p_0 = 20 and t=100t = 100: p100=20+1000.13p_{100} = 20 + 100 \cdot 0.13

Computing p100p_{100}:

Let’s calculate: p100=20+1000.13p_{100} = 20 + 100 \cdot 0.13

I will now compute both results. Part (a) calculation p0_a = 14 factor_a = 0.96 t_a = 69

p_69 = p0_a * (factor_a ** t_a)

Part (b) calculation

p0_b = 20 increment_b = 0.13 t_b = 100

p_100 = p0_b + t_b * increment_b

p_69, p_100The solutions are as follows:

Part (a):

p690.8372p_{69} \approx 0.8372

Part (b):

p100=33.0p_{100} = 33.0

Would you like a detailed breakdown of the steps?

Here are five related questions to expand on this topic:

  1. How does the solution change if the recurrence relations are modified slightly (e.g., different constants)?
  2. What are some real-world applications of recurrence relations like these?
  3. Can you derive the formulas for geometric and arithmetic progressions from scratch?
  4. How can recurrence relations be solved for non-linear terms?
  5. What tools or software can help in solving more complex recurrence relations?

Tip: Always double-check your initial conditions, as they are critical in determining accurate results in recurrence problems.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Recurrence Relations
Geometric Progressions
Arithmetic Progressions

Formulas

p_t = p_0 * (r)^t (for geometric progression)
p_t = p_0 + t * d (for arithmetic progression)

Theorems

Properties of geometric progressions
Properties of arithmetic progressions

Suitable Grade Level

Grades 9-12