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Sequences and Series

Short Answer Type

111. If a, b, c, d are in GP., show that (a² + b² + c²) (b² + c² + d²) = (ab + bc + cd)²

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Long Answer Type

112.

If a, b, c are in G.P. and x, y are the arithmetic means of a, b and b, c respectively,

then prove that $straight a over straight x plus straight c over straight y equals 2$ and $space space 1 over straight x plus 1 over straight y equals 2 over straight b$

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Short Answer Type

113.

If a, b, c are in A.P. and x,y,z are in G.P., show that

$straight x to the power of straight b minus straight c end exponent. space straight y to the power of straight c minus straight a end exponent. space straight z to the power of straight a minus straight b end exponent space equals space 1$

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114. Find a GP. for which sum of first two terms is –4 and fifth term is 4 times the 3rd term.

Let the G.P. be a, ar, ar², ..........

Given a + ar = -4 ...(i)

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or $space space space ar to the power of 5 minus 1 end exponent equals space 4. space ar to the power of 3 minus 1 end exponent space rightwards double arrow space ar to the power of 4 minus 4 ar squared space equals space 0 space rightwards double arrow space ar squared left parenthesis straight r squared minus 4 right parenthesis space equals space 0$

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

$rightwards double arrow$ $straight r equals plus-or-minus 2.$

Putting r = 2 in (i), we get a + 2a = -4 $rightwards double arrow$ a = -4/3.

∴ G.P. is $space space space fraction numerator negative 4 over denominator 3 end fraction comma space fraction numerator negative 4 over denominator 3 end fraction left parenthesis 2 right parenthesis. space fraction numerator negative 4 over denominator 3 end fraction left parenthesis 2 right parenthesis squared comma space.......... space or space fraction numerator negative 4 over denominator 3 end fraction comma space fraction numerator negative 8 over denominator 3 end fraction comma space fraction numerator negative 16 over denominator 3 end fraction comma space.........$

Now, putting r = -2 in (i), we get a + a(-2) = -4 $rightwards double arrow$ -a = -4 or a = 4.

∴ G.P. is 4, 4(-2), 4(-2)², .... or 4, -8, 16, ...........