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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.

Since A is the A.M. between a and b

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

Since G is the G.M. between a and b

∴ G = $space space square root of ab$ ...(ii)

A - G = $fraction numerator straight a plus straight b over denominator 2 end fraction minus square root of ab space equals space fraction numerator straight a plus straight b minus 2 square root of ab over denominator 2 end fraction equals space fraction numerator left parenthesis square root of straight a right parenthesis squared plus left parenthesis square root of straight b right parenthesis squared minus 2 square root of straight a square root of straight b over denominator 2 end fraction space equals space fraction numerator left parenthesis square root of straight a minus square root of straight b right parenthesis squared over denominator 2 end fraction greater than 0$

∴ A - G > 0 $rightwards double arrow$ 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.

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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

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