Two functions to add CPT archives into the mix for post archives and front page loops.

add_cpt_to_posts_archives.php

add_filter( 'pre_get_posts', 'prefix_cpk_show_cpt_archives' );
add_action( 'pre_get_posts', 'prefix_cpk_add_custom_post_types_to_loop' );
/**
 * prefix_cpk_show_cpt_archives - Display job posting entires on taxonomy archive pages
 *
 * @param [type] $query
 * @return void
 */
function prefix_cpk_show_cpt_archives( $query ) {
	if( is_category() || is_tag() && empty( $query->query_vars['suppress_filters'] ) ) { 
		$query->set( 'post_type', array(
			'post', 'nav_menu_item', 'job_posting'
		));
		return $query;
	}
}

/**
 * prefix_cpk_add_custom_post_types_to_loop - Add JobPosting CPT to front page loop.
 *
 * @param [type] $query
 * @return void
 */
function prefix_cpk_add_custom_post_types_to_loop( $query ) {
	if ( is_home() && $query->is_main_query() ) {
		$query->set( 'post_type', array( 'post', 'job_posting' ) );
		return $query;
    }
}