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Sequences and Series

Short Answer Type

61.

The fourth term of a GP. is 27 and the 7th term is 729, find the GP.

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62.

The seventh term of a GP. is 8 times the fourth term and 5th term is 48. Find the GP.

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63.

If the GP.’s 5,10, 20, .......and 1280, 640, 320, ......have their nth terms equal, find the value of n.

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64.

The third term of a GP. is 4. Find the product of its first five terms.

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65.

The 5th, 8th and 11th terms of a GP. are p, q and s respectively. Show that q² = ps.

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66.

Find all the sequences which are simultaneously in A.P. and GP.

343 Views

67.

In a finite GP. the product of the terms equidistant from the beginning and the end is always same and equal to the product of first and last term.

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68.
If the first and the nth terms of a GP. are a and b respectively and if P is the product of first n terms, prove that P² = (ab)ⁿ.

Here, a is the first term, let r be the common ratio.

Also, $straight t subscript straight n space equals space straight b space space space rightwards double arrow space space ar to the power of straight n minus 1 end exponent space equals space straight b$

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

Since P is the product of first n terms

∴ $straight P space equals space straight t subscript 1. space straight t subscript 2. space straight t subscript 3 space................ space straight t subscript straight n space equals space straight a. ar. ar squared. space............... space space space ar to the power of straight n minus 1 end exponent$ $open curly brackets because space space In space straight A. straight P. space straight S subscript straight n space equals space straight n over 2 left square bracket straight a plus straight l right square bracket close curly brackets$

 $equals space straight a to the power of straight n straight r to the power of 1 plus 2 plus.......... plus space left parenthesis straight n space minus space 1 right parenthesis end exponent space equals space straight a to the power of straight n space. straight r to the power of fraction numerator straight n minus 1 over denominator 2 end fraction left parenthesis 1 plus straight n minus 1 right parenthesis end exponent$

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

 $equals straight a to the power of straight n open square brackets open parentheses straight b over straight a close parentheses to the power of fraction numerator 1 over denominator straight n minus 1 end fraction end exponent close square brackets to the power of fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 end fraction end exponent space equals space straight a to the power of straight n open parentheses straight b over straight a close parentheses to the power of straight n divided by 2 end exponent space equals space straight a to the power of straight n. space straight b to the power of straight n divided by 2 end exponent over straight a to the power of straight n divided by 2 end exponent space equals space straight a to the power of straight n minus straight n over 2 end exponent straight b to the power of straight n over 2 end exponent equals space straight a to the power of straight n over 2 end exponent straight b to the power of straight n over 2 end exponent$

$rightwards double arrow$ $space space straight P space equals space left parenthesis ab right parenthesis to the power of straight n over 2 end exponent$

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

107 Views

Long Answer Type

69.

If $fraction numerator straight a plus bx over denominator straight a minus bx end fraction space equals space fraction numerator straight b plus cx over denominator straight b minus cx end fraction space equals space fraction numerator straight c plus dx over denominator straight c minus dx end fraction$ $left parenthesis straight x not equal to 0 right parenthesis comma space$ show that a, b, c, d are in G.P.