Number of posts in the archive

You can change the posts_per_page variable depending on the page you are on, so make it 10 on the first page and 20 on the other pages. However, you will have to modify the offset query variable too, otherwise you will skip posts 11-20 on the second page, because this page thinks they are already displayed on the first page.

add_action( 'pre_get_posts', 'wpse15087_pre_get_posts' );
function wpse15087_pre_get_posts( &$wp_query )
{
    if ( $wp_query->is_post_type_archive( 'apartments' ) ) {
        if ( ! $wp_query->is_paged() ) {
            $wp_query->set( 'posts_per_page', 10 );
        } else {
            $wp_query->set( 'posts_per_page', 20 );
            $wp_query->set( 'offset', ( ( $wp_query->get( 'paged' ) - 2 ) * 20 ) + 10 );
        }
    }
}