1. please set up recurrence equation

Let an be the number of subsets of {1,2,3,..n} with atleast 2 elements,n>=1.
i)what are the values of a1,a2,a3?
2)set up a recuurence for an and solve it.

2. Hello, Bhavani!

Let $a_n$ be the number of subsets of $\{1,2,3,\hdots n\}$

1) What are the values of: $a_1,\:a_2,\:a_3$ ?

For $n = 1$, we have: . $S \:=\:\{1\}$
There are two subsets: . $\emptyset,\:\{1\}$
. . $a_1 = 2$

For $n=2$, we have: . $S \:=\:\{1,2\}$
There are four subsets: . $\emptyset,\:\{1\},\:\{2\},\:\{1,2\}$
. . $a_2 = 4$

For $n=3$, we have: . $S \:=\:\{1,2,3\}$
There are eight subsets: . $\emptyset,\:\{1\}, \{2\}, \{3\}, \{1,2\}, \{2,3\}, \{1,3\}, \{1,2,3\}$
. . $a_3 = 8$

2) Set up a recurrence for $a_n$ and solve it.

The recurrence is: . $a_n \;=\;2\cdot a_{n-1},\quad a_1 = 2$

The closed form is: . $a_n \:=\:2^n$

3. hello soroban,