Так объяви в самом начале:

$rand = array();
$fullRand = '';

<?php
session_start();

$image = imageCreate(200, 50);

$bg = imagecolorallocate($image, 255, 255, 255);

$abc = array('A','B','C','D','E','F','G','H','I','J','K', 'L','M','N','1','2','3','4','5','6','7','8','9');

for ($i=0;$i<6;$i++) {
$rand[$i] = $abc[mt_rand(0,22)];
}

for ($i=0;$i<6;$i++) {
$textcolor = imagecolorallocate($image, rand(0,255), rand(0,255), rand(0,255));
imagettftext( $image, 20, 0 + 10*$i, 10 + 25*$i, 25, $textcolor, "./Arial.ttf", $rand[$i]);
imageArc($image, rand(0,90), rand(0,20), rand(10,60), rand(10,60), rand(-380,0), rand(0,380), $textcolor);
}

$_SESSION['RAND_STRING'] = implode("",$rand);

header('Content-type: image/png');

imageInterlace($image, 1);

imageColorTransparent($image, $colorBackgr);

imagePng($image);
?>