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

$2 straight A space equals space fraction numerator straight g subscript 1 superscript 2 over denominator straight g subscript 2 end fraction space plus space fraction numerator straight g subscript 2 superscript 2 over denominator straight g subscript 1 end fraction$

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102.

If a be the A.M. and x, y be the two G.M.'s between b and c, show that

$straight x cubed plus straight y cubed space equals space 2 abc$

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103.

The sum of first three terms of a G.P. is $13 over 12$ and their product is -1. Find the terms.

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104.

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is $87 1 half.$ Find them.

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Long Answer Type

105.

The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.

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Short Answer Type

106. Find four numbers forming a GP. in which the third term is greater than the first by 9 and the second term is greater than the fourth by 18.

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Long Answer Type

107. The sum of three numbers in GP. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.

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Short Answer Type

108. The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a GP. Find the numbers.

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109. If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that (p – q), (q – r), (r – s) are in GP.

Let A be the first term and D be the common difference.

∴ $straight t subscript straight p space equals space straight A plus left parenthesis straight p minus 1 right parenthesis straight D comma space space space space straight t subscript straight q space equals space straight A plus left parenthesis straight q minus 1 right parenthesis straight D comma space space straight t subscript straight r space equals space straight A plus left parenthesis straight r minus 1 right parenthesis straight D comma$

 $straight t subscript straight s space equals space straight A plus left parenthesis straight s minus 1 right parenthesis straight D$

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

$straight t subscript straight q minus straight t subscript straight r space equals space left parenthesis straight q minus straight r right parenthesis straight D$ ...(ii)

 $straight t subscript straight r minus straight t subscript straight s space equals space left parenthesis straight r minus straight s right parenthesis straight D$ ...(iii)

Since $straight t subscript straight p comma space straight t subscript straight q comma space straight t subscript straight r comma space straight t subscript straight s$ are in G.P.

∴ $straight t subscript straight q over straight t subscript straight p space equals space straight t subscript straight r over straight t subscript straight q space equals space straight t subscript straight s over straight t subscript straight r space equals space space straight R left parenthesis say right parenthesis comma space space straight t subscript straight q equals straight t subscript straight p straight R comma space straight t subscript straight r space equals space straight t subscript straight q straight R space equals straight t subscript straight p straight R squared comma space straight t subscript straight s equals straight t subscript straight r straight R space equals space straight t subscript straight p straight R cubed$

Now, $space space straight t subscript straight q over straight t subscript straight p equals space straight t subscript straight r over straight t subscript straight q$ and $straight t subscript straight r over straight t subscript straight q space equals space straight t subscript straight s over straight t subscript straight r$

$rightwards double arrow$ $straight t subscript straight q over straight t subscript straight p minus 1 space equals space straight t subscript straight r over straight t subscript straight q minus 1$ and $straight t subscript straight r over straight t subscript straight q minus 1 space equals space straight t subscript straight s over straight t subscript straight r minus 1$

$rightwards double arrow$ $fraction numerator straight t subscript straight q minus straight t subscript straight p over denominator straight t subscript straight p end fraction space equals space fraction numerator straight t subscript straight r minus straight t subscript straight q over denominator straight t subscript straight p straight R end fraction$ and $space space fraction numerator straight t subscript straight r minus straight t subscript straight q over denominator straight t subscript straight p straight R end fraction space equals space fraction numerator straight t subscript straight s minus straight t subscript straight r over denominator straight t subscript straight p straight R squared end fraction$

$rightwards double arrow$ $fraction numerator negative left parenthesis straight p minus straight q right parenthesis straight D over denominator 1 end fraction space equals space fraction numerator negative left parenthesis straight q minus straight r right parenthesis straight D over denominator straight R end fraction$ and $fraction numerator negative left parenthesis straight q minus straight r right parenthesis straight D over denominator 1 end fraction space equals space fraction numerator negative left parenthesis straight r minus straight s right parenthesis straight D over denominator straight R end fraction$

$rightwards double arrow$ $fraction numerator straight q minus straight r over denominator straight p minus straight q end fraction space equals space straight R$ and $fraction numerator straight r minus straight s over denominator straight q minus straight r end fraction space equals space straight R$

$rightwards double arrow$ $space space fraction numerator straight q minus straight r over denominator straight p minus straight q end fraction space equals space fraction numerator straight r minus straight s over denominator straight q minus straight r end fraction$

$rightwards double arrow$ p - q, q - r, r - s are in G.P.

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110. If a, b, c are in GP., prove that a² + b², ab + bc, b² + c² are also in GP.

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