Question

Let $S=\{4,6,9\}$ and $T=\{9,10,11, \ldots$, 1000 . If $A =\left\{ a _1+ a _2+\ldots+ a _{ k }: k \in N , a _1, a _2\right.$, $\left.a_3, \ldots, a_k \in S\right\}$, then the sum of all the elements in the set $T - A$ is equal to

Answer 1

$ S =\{4,6,9\} \quad T =\{9,10,11 \ldots .1000\} $
$A \left\{ a _1+ a _2+\ldots . .+ a _{ k }: K \in N \right\} \& a _{ i } \in S$
Here by the definition of set 'A'
$A=\{a: a=4 x+6 y+9 z\}$
Except the element 11, every element of set $T$ is of the form $4 x+6 y+9 z$ for some
$ x , y , z \in W$
$\therefore T - A =\{11\}$