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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.

Let a, ar, ar² be three numbers in G.P.

Since their sum is 56

∴ a + ar + ar² = 56

$rightwards double arrow$ a(1 + r + r²) = 56 ...(i)

It is given that if we subtract 1, 7, 21 from the numbers respectively then the resulting numbers form an A.P.

∴ a - 1, ar - 7, ar²- 21 are in A.P.

$rightwards double arrow space space space space space space space space left parenthesis ar minus 7 right parenthesis space minus space left parenthesis straight a minus 1 right parenthesis space equals space left parenthesis ar squared minus 21 right parenthesis space minus space left parenthesis ar minus 7 right parenthesis$ $rightwards double arrow space space space ar minus 7 minus straight a plus 1 space equals space ar squared space minus 21 minus ar plus 7$

$rightwards double arrow$ $ar squared minus 2 ar plus straight a equals 8$ $rightwards double arrow$ $straight a left parenthesis straight r squared minus 2 straight r plus 1 right parenthesis space equals space 8$ ...(ii)

Dividing (i) by (ii), we get $fraction numerator straight a left parenthesis straight r squared plus straight r plus 1 right parenthesis over denominator straight a left parenthesis straight r squared minus 2 straight r plus 1 right parenthesis end fraction space equals space 56 over 8$

$rightwards double arrow$ $straight r squared plus straight r plus 1 space equals space 7 left parenthesis straight r squared minus 2 straight r plus 1 right parenthesis space rightwards double arrow space space 6 straight r squared minus 15 straight r plus 6 space equals space 0 space space rightwards double arrow space 2 straight r squared minus 5 straight r plus 2 space equals space 0$

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

when r = 2, from (i), $space space space straight a left parenthesis 1 plus 2 plus 2 squared right parenthesis space equals space 56 space rightwards double arrow space space 7 straight a space equals space 56 space space rightwards double arrow space space straight a space equals space 8$

∴ Numbers are 8, 8(2), 8(2)² or 8, 16, 32

When $straight r equals space 1 half$ from (i), $straight a open parentheses 1 plus 1 half plus 1 fourth close parentheses space equals space 56 space space space rightwards double arrow space space straight a open parentheses 7 over 4 close parentheses space equals space 56 space space space space rightwards double arrow space space straight a space equals space 32$

∴ Numbers are $32 comma space 32 open parentheses 1 half close parentheses comma space 32 open parentheses 1 half close parentheses squared$ or 32, 16, 8

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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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