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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.

Let a, ar, ar², ar³ be the four numbers in G.P.

Since the third number is greater than the first by 9

∴ ar² = a + 9    $rightwards double arrow$ a(r² - 1) = 9 ...(i)

Since the second number is greater than the fourth by 18

∴ ar = ar³ + 18 $rightwards double arrow$ ar(r² - 1) = -18 ...(ii)

Dividing (ii) by (i), we get

   $fraction numerator ar left parenthesis straight r squared minus 1 right parenthesis over denominator straight a left parenthesis straight r squared minus 1 right parenthesis end fraction space equals space fraction numerator negative 18 over denominator 9 end fraction space space space space space space space space space space space space rightwards double arrow space space straight r space equals space minus 2$

Using r = -2 in (i), we get

   $straight a left square bracket left parenthesis negative 2 right parenthesis squared minus 1 right square bracket space equals space 9$ $rightwards double arrow$ a(4 - 1) = 9 $rightwards double arrow$ a = 3

∴ Numbers are 3, 3(-2), 3(-2)², 3(-2)³ or 3, -6, 12, -24

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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.

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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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