Question

Give a recursive definition for the set X of all natural numbers that are one or two more than a multiple of 10. In other words, give a recursive definition for the set {1, 2, 11, 12, 21, 22, 31, 32, ... }.
B1. 1 is in X.
B2. __ is in X.
R. If x is in X, so is ____

Answer 1

I know you are not here, still give you the answer. Hey, !! it's tricky Ma'am!!
to me, you must divide the set into 2 parts. I label the term as \[a_1,a_2,a_3.....\]
and separate them as \[a_1, a_3, a_5.....\]and \[ {a_2, a_4, a_6....
since they work on the same but the initial condition is different.
Really not know whether it makes sense to you or not. just continue,

Answer 2

for part 1, \[a_1 = 1\]
\[a_3 = a_1 +10 \]
\[a_5 = a_1 + 2*10\]
\[a_7 = a_1 + 3 *10\]
and so on......
you can conclude that \[a_n = a_1 + (\frac{n-1}{2})*10\]

Answer 3

for part 2 the same logic when constructing the recursive relation
\[a_2=2\]
\[a_4 = 2 + 10=a_2 + 10\]
\[a_6 = 2 + 2*10= a_2 + 2*10\]
\[a_8 = 2 + 2*10= a_2 + 3*10\]
so on..... you can conclude that \[a_n= a_2 + (\frac{n-2}{2})*10\]

Answer 4

from then, when combine them, you should give out the conclusion in the form of