Question

A sequence is defined recursively by s_n= -s_(n-1)+3, for all integers n≥1,s_0=1. Write down the first four elements and develop an explicit formula for the sequence.

Answer 1

The given sequence is defined recursively as s_n = -s_(n-1) + 3. The first four elements of the sequence are s_0 = 1, s_1 = 2, s_2 = 1, and s_3 = 2. The explicit formula for the sequence is s_n = 1 + (-1)^n.

Step-by-step explanation:

The given sequence is defined recursively as:

sn = -sn-1 + 3

To find the first four elements, we can start with the initial condition s0 = 1 and use the recursive formula:

1. s1 = -s0 + 3 = -1 + 3 = 2
2. s2 = -s1 + 3 = -2 + 3 = 1
3. s3 = -s2 + 3 = -1 + 3 = 2
4. s4 = -s3 + 3 = -2 + 3 = 1

From the given calculations, the first four elements of the sequence are:

s0 = 1, s1 = 2, s2 = 1, s3 = 2

To develop an explicit formula for the sequence, we can observe that the sequence alternates between 1 and 2. When n is even, sn = 2, and when n is odd, sn = 1. This can be represented using the formula:

sn = 1 + (-1)n