Let S={4,6,9} and T={9,10,11,.....,1000}. If A={a1+a2+......+ak:k∈N,a1,a2,a3,.....,ak∈S}, then twice the sum of all the elements in the set T−A is equal to___________.

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Let $S = {4, 6, 9}$ and $T = {9, 10, 11, ....., 1000} .$ If $A = {a_{1} + a_{2} + ...... + a_{k} : k \in N, a_{1}, a_{2}, a_{3}, ....., a_{k} \in S},$ then twice the sum of all the elements in the set $T - A$ is equal to___________.

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Given $S = {4, 6, 9}$ and $T = {9, 10, 11, ....., 1000}$

Where, $A = {a_{1} + a_{2} + ...... + a_{k} : K \in N} & a_{i} \in S$

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 $x, y, z \in W$

$∴ T - A = {11}$

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