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$space space sum from straight i equals 1 to 10 of open square brackets open parentheses 1 half close parentheses to the power of straight i minus 1 end exponent plus open parentheses 1 fifth close parentheses to the power of straight i plus 1 end exponent close square brackets$

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92.

Find the sum of the products of the corresponding terms of sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2, $1 half.$

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93.

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

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94.

If A and G are respectively the A.M. and G.M. between the two distinct positive numbers are a and b, then Prove A>G.

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95.

If A and G are respectively the A.M. and G.M. between two positive numbers a and b, then proof the quardratic equation having a, b as it roots is $straight x squared minus 2 Ax plus straight G squared equals 0.$

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96.

If A and G be A.M. and G.M. between two positive numbers, then the numbers are

$straight A plus-or-minus square root of straight A squared minus straight G squared end root.$

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97.

The A.M. between two positive numbers is 34 and their GM. is 16. Find the numbers.

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98.
Find two positive numbers whose difference is 12 and whose A.M. exceeds the GM. by 2.

Let a and b be two positive numbers such that

a - b = 12 ...(i)

Since A.M. exceeds G.M. by 2

∴ $fraction numerator straight a plus straight b over denominator 2 end fraction space equals space square root of ab plus 2$ $rightwards double arrow$ a + b = $2 square root of ab plus 4$

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

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

From (i), $left parenthesis square root of straight a minus square root of straight b right parenthesis end root space left parenthesis square root of straight a plus square root of straight b right parenthesis space equals space 12 space rightwards double arrow space space 2 left parenthesis square root of straight a plus square root of straight b right parenthesis space equals space 12 space rightwards double arrow space space square root of straight a plus square root of straight b space equals space 6$ ...(iii)

Adding (ii) and (iii), we get $2 square root of straight a space equals space 8 comma space space space square root of straight a space equals space 4 space space space rightwards double arrow space space space straight a space equals space 16$

Subtracting (ii) from (iii), we get $2 square root of straight b space equals space 4 space rightwards double arrow space space square root of straight b space equals space 2 space rightwards double arrow space straight b space equals space 4$

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99.

Find the value of n so that $fraction numerator straight a to the power of straight n plus 1 end exponent plus straight b to the power of straight n plus 1 end exponent over denominator straight a to the power of straight n plus straight b to the power of straight n end fraction$ may be the G.M. between a and b.

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100.

The A.M. of two numbers is 3 times their G.M. Show that the numbers are in the ratio

$3 plus 2 square root of 2 colon space space 3 minus 2 square root of 2.$