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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

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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.

Let $space space straight S subscript straight n space equals space 1 plus 3 plus 3 squared plus...... space to space straight n space terms$

Here, a = 1, r = 3

$straight S subscript straight n. space space space space fraction numerator straight a left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction space equals space fraction numerator 1 left parenthesis 3 to the power of straight n minus 1 right parenthesis over denominator 3 minus 1 end fraction space equals space fraction numerator 3 to the power of straight n minus 1 over denominator 2 end fraction$

Since $straight S subscript straight n greater than 7000$

∴ $fraction numerator 3 to the power of straight n minus 1 over denominator 2 end fraction greater than 7000$ $rightwards double arrow space space space space space space space space space space space space space space space space space 3 to the power of straight n minus 1 space greater than space 14000$ $rightwards double arrow space space space space space space space space space space space 3 to the power of straight n greater than 14001$

$rightwards double arrow$ $3 to the power of straight n space greater than space 19683 space greater than space 14001 space greater than space 6561$

$rightwards double arrow$ $3 to the power of straight n greater than space 3 to the power of 9 greater than 14001 space greater than space 3 to the power of 8$

Hence, the least value of n is 9.

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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:

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