A circular disk is cut into n distinct scctuis. each shaped like a piece of pie and all meeting at the center point of the disk. Each sector is to be painted red. green, yellow, or blue in such a way that no two adjacent sectors arc panned tlx: vainc coJoi. Let .S. be tlie nunibei of ways to paint the disk.
a. Find a recurrence retains Cor S_kia terms ofand for each integer 12 4.
b. Find an explicit lor inula for S. for n >2L

(a)

To find the recurrence relation for Sk in terms of Sk−1 and Sk−2 for each integer k≥4

Given information:

A circular disk is cut into n distinct sectors. No two adjacent sectors are of the same color because each adjacent sector in painted with different color in the circular disk. The four colors are red, green and yellow or blue.

Calculation:

Let Sk be the number of ways to paint the circular disk.

Since there are four colors so if suppose one color say red is used for painting the sector, so adjacent sector will be left with 3 choices of colors to paint either green, yellow or blue. So all the sectors will have three choices except the last one which have two neighbor on its adjacent sides

(b)

To find the explicit formula for n≥2.