As a simplified model for weather forecasting, suppose that the weather (either wet or dry) tomorrow will be the same as the weather today with probability p. show that the weather is dry on january 1, then

pn

, the probability that it will be dry n days layer, satisfies

pn=(2p−1)pn−1+(1−p)n≥1

p0=1

prove that

pn=1/2+1/2(2p−1)nn≥0

A3cm×3cm\greend{3\,\text{cm} \times 3\,\text{cm}} 3cm×3cm start color greend, 3, space, c, m, times, 3, space, c, m, end color greend rectangle sits inside a circle with radius of 4cm\blued{4\,\text{cm}} 4cm start color blued, 4, space, c, m, end color blued . what is the area of the shaded region? round your final answer to the nearest hundredth.