smallest number in a sequence

Let $a$ be the middle term and $d$ the common difference, then the progression is $a-d,a,a+d$ .

So we have

$a-d+a+a+d=66$ $\color{#D61F06}\implies$ $a=22$ .

The three terms are $22-d,22,22+d$ . Therefore,

$(22-d)^2+22^2+(22+d)^2=1694$ $\color{#D61F06}\implies$ $22^2-44d+d^2+22^2+22^2+44d+d^2=1694$ $\color{#D61F06}\implies$ $1452+2d^2=1694$ $\color{#D61F06}\implies$ $d^2=121$ $\color{#D61F06}\implies$ $d=\pm 11$

If $d=11$ , the terms are $11,22,33$ .

If $d=-11$ , the terms are $33,22,11$ .

Thus, the smallest number in this progression is $\boxed{11}$ .