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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.

Given                           f(1) = 3

and                          f(x + y) = f(x). f(y) ...(ii)           $horizontal strike logical or$ x, $straight y element of straight N.$ ...(i)

Putting x = 1, y = 1 in (ii), we get

                               f(1 + 1) = f(1). f(1) $rightwards double arrow$ f(2) = 3. 3 = 9.

Putting x = 1, y = 2 in (ii), we get

                              f(1 + 2) = f(1). f(2) = 3 . 9 = 27 or f(3) = 27

Putting x = 1, y = 3 in (ii), we get

                             f(1 + 3) = f(1). f(3) = 3. 27 = 81 or f(4) = 81

∴ f(1) = 3, f(2) = 9, f(3) = 27, f(4) = 81, ...............

Here, the sequence 3, 9, 27, 81, .................. is a G.P. with a = 3, r = 3.

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

or 3 + 9 + 27 +............ to n terms = 120

$rightwards double arrow$ $fraction numerator 3 left parenthesis 3 to the power of straight n minus 1 right parenthesis over denominator 3 minus 1 end fraction space equals space 120 space rightwards double arrow space 3 space left parenthesis 3 to the power of straight n minus 1 right parenthesis space equals space 240 space rightwards double arrow space 3 to the power of straight n minus 1 space equals space 80 space space space rightwards double arrow space space 3 to the power of straight n space equals space 81 space equals space 3 to the power of 4$

∴ n = 4

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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.

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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

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