Find n, if the ratio of the fifth term from the beginning to the

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Binomial Theorem

Short Answer Type

71.

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

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72.

Find the 11^th term from the end in the expansion of $open parentheses 2 straight x minus 1 over straight x squared close parentheses to the power of 25$

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73. Prove that in the expansion of (a + b)ⁿ, the co-efficients of terms equidistant from the beginning and end are the same.

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Long Answer Type

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74.
Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $open parentheses fourth root of 2 plus fraction numerator 1 over denominator fourth root of 3 end fraction close parentheses to the power of straight n$ is $square root of 6 colon 1.$

open parentheses fourth root of 2 plus fraction numerator 1 over denominator fourth root of 3 end fraction close parentheses to the power of straight n

space 5 to the power of th space

term from the beginning =

space space straight T subscript 5 equals straight C presuperscript straight n subscript 4 space left parenthesis fourth root of 2 right parenthesis to the power of straight n minus 4 end exponent. space open parentheses fraction numerator 1 over denominator fourth root of 3 end fraction close parentheses to the power of 4

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from the end =

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Ratio of the two terms is

straight C presuperscript straight n subscript 4 space left parenthesis fourth root of 2 right parenthesis to the power of straight n minus 4 end exponent space fraction numerator 1 over denominator left parenthesis fourth root of 3 right parenthesis to the power of 4 end fraction space colon space straight C presuperscript straight n subscript straight n minus 4 end subscript space left parenthesis fourth root of 2 right parenthesis to the power of 4 space fraction numerator 1 over denominator left parenthesis fourth root of 3 right parenthesis to the power of straight n minus 4 end exponent end fraction

But

space space straight C presuperscript straight n subscript 4 equals straight C presuperscript straight n subscript straight n minus 4 end subscript

∴ The ratio is

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Hence, by the given condition,

space space space space fraction numerator left parenthesis fourth root of 2 right parenthesis to the power of straight n minus 4 end exponent over denominator left parenthesis fourth root of 3 right parenthesis to the power of 4 end fraction cross times fraction numerator left parenthesis fourth root of 3 right parenthesis to the power of straight n minus 4 end exponent over denominator left parenthesis fourth root of 2 right parenthesis to the power of 4 end fraction equals fraction numerator square root of 6 over denominator 1 end fraction space space rightwards double arrow space space left parenthesis fourth root of 2 right parenthesis to the power of straight n minus 8 end exponent. space left parenthesis fourth root of 3 right parenthesis to the power of straight n minus 8 end exponent space equals space square root of 6

rightwards double arrow

space space left parenthesis 2 right parenthesis to the power of fraction numerator straight n minus 8 over denominator 4 end fraction end exponent space. space left parenthesis 3 right parenthesis to the power of fraction numerator straight n minus 8 over denominator 4 end fraction end exponent space equals space 6 to the power of 1 half end exponent

rightwards double arrow

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rightwards double arrow

space space fraction numerator straight n minus 8 over denominator 4 end fraction space equals space 1 half

rightwards double arrow

n - 8 = 2

Hence, n = 10

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Short Answer Type

75. Find the co-efficient of

straight x to the power of 5 space in space left parenthesis straight x plus 3 right parenthesis to the power of 8

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76. Find the co-efficient of:

straight a to the power of 5 straight b to the power of 7 space in space left parenthesis straight a minus 2 straight b right parenthesis to the power of 12

.

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77. Find the positive value of m for which the co-efficient of x² in the expansion of (1 + x)^m is 6.

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78. In the expansion of (1 + x)ⁿ, the co-efficients of x^p-1 and of x^q-1 are equal. Show that p + q = n + 2, p ≠ q.

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79. In the expansion of (1 + a)^m+n , prove that co-efficient of a^m and aⁿ are equal.

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80. Find a, if the co-efficients of x² and x³ in the expansion of (3 + ax)⁹ are equal.

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7
8
9

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