Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0000 362584 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0502 1212056 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0502 1212056 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1672 4127672 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1685 4155352 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1692 4173064 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1694 4203584 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1704 4226304 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1704 4226304 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2251 4896256 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2287 5088408 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2293 5109336 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2403 5318984 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2404 5319240 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Total Area of Concentric Annuli With Radii in Arithmetic Sequence

Suppose we have an arrangement of spherical concentric annuluses, each of thickness 1, separated by spaces of thickness 1.
The first annulus has outer thickness 1 and inner thickness 0. The second annulus has outer thickness 3 and inner thickness 2. The third annulus has outer thickness 5 and inner thickness 4. Continue in this way, then the k<supth annulus has outer radius  

\[2k-1\]
  and inner radius  
\[2k-2\]

The kth annulus will have area  
\[\pi ((2k-1)^2 - (2k-2)^2) = \pi ( 4k-3) \]

Adding up the areas of these  
\[n\]
  shells gives
\[\begin{equation} \begin{aligned} A &= \pi \sum^n_1 4k-3 \\ &= 4 \pi \sum^n_1 K - 3 \pi \sum 1 \\ &= 2 \pi n(n+1) - 3 \pi n \\ &= \pi n( 2n-1) \end{aligned} \end{equation}\]

on using using these identities