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How many terms of the sequence

$space space square root of 3 comma space 3 comma space 3 square root of 3 comma space........$ must be taken to make the sum $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ ?

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82.

Sum the following series:

7 + 77 + 777 + ......... to n terms

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83.
Sum the following series:

0.5 + 0.55 + 0.555.......... to n terms

Let $space space straight S subscript straight n$ = 0.5 + 0.55 + 0.555 +............ to n terms

           = 5[0.1 + 0.11 + 0.111 + ........... to n terms]

           = $5 over 9 left square bracket 0.9 space plus space 0.99 space plus space 0.999 space plus space.......... space to space straight n space terms right square bracket$

           = $5 over 9 left square bracket left parenthesis 1 minus 0.1 right parenthesis space plus space left parenthesis 1 minus.01 right parenthesis space plus space left parenthesis 1 minus.001 right parenthesis space plus space....... space to space straight n space terms right square bracket$

           = $5 over 9 open square brackets straight n minus open parentheses 1 over 10 plus 1 over 100 plus 1 over 1000 plus....... space to space straight n space terms close parentheses close square brackets$

           = $5 over 9 open square brackets straight n minus fraction numerator begin display style 1 over 10 end style open parentheses 1 minus begin display style 1 over 10 to the power of straight n end style close parentheses over denominator 1 minus begin display style 1 over 10 end style end fraction close square brackets space equals space 5 over 9 open square brackets straight n minus 1 over 9 open parentheses 1 minus 1 over 10 to the power of straight n close parentheses close square brackets space equals space 5 over 81 open square brackets 9 straight n minus 1 space plus space 1 over 10 to the power of straight n close square brackets$

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84.

Given a G.P. with a = 729 and 7th term 64, determine S₇.

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85.

Determine the number of terms in a GP., if t₁ = 3, t_n = 96 and S_n = 189.

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86.

Find the least value of n for which the sum 1 + 3 + 3²+ ...... to n terms is greater than 7000.

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87.

The ratio of the sum of first three terms is to that of first 6 terms of a G.P. is 125 : 152. Find the common ratio.

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88.

If S₁, S₂ and S₃ be respectively the sums of n, 2n and 3n terms of a GP., prove that S₁ (S₃ – S₂) = (S₂ – S₁)²

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Long Answer Type

89.

Let S be the sum, P the product and R the sum of the reciprocals of n terms in a G.P. Prove that $straight P squared straight R to the power of straight n space equals space straight S to the power of straight n.$

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Short Answer Type

90.

Evaluate:

$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$