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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

1078 Views

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.

321 Views

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.

Let a be the first term and r be the common ratio

According to the given equation,

$straight S subscript 3 over straight S subscript 6 space equals space 125 over 152 space space space space space space space rightwards double arrow space space space space space fraction numerator begin display style fraction numerator straight a left parenthesis straight r cubed minus 1 right parenthesis over denominator straight r minus 1 end fraction end style over denominator begin display style fraction numerator straight a left parenthesis straight r to the power of 6 minus 1 right parenthesis over denominator straight r minus 1 end fraction end style end fraction space equals space 125 over 152 space space space space space space space space space space space rightwards double arrow space space space space fraction numerator straight r cubed minus 1 over denominator straight r to the power of 6 minus 1 end fraction space equals space 125 over 152$

$rightwards double arrow space fraction numerator straight r cubed minus 1 over denominator left parenthesis straight r cubed minus 1 right parenthesis left parenthesis straight r cubed plus 1 right parenthesis end fraction space equals space 125 over 152 space space space space rightwards double arrow space space space fraction numerator 1 over denominator straight r cubed plus 1 end fraction space equals space 125 over 152 space space space space rightwards double arrow space space space space 125 straight r cubed space plus space 125 space equals space 152$

$rightwards double arrow$ $125 space straight r cubed space equals space 27$ $rightwards double arrow space space space space space space space space straight r cubed space equals space 27 over 125$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

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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

1051 Views

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 $straight P squared straight R to the power of straight n space equals space straight S to the power of straight n.$

131 Views

Short Answer Type

90.

Evaluate:

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