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Principle of Mathematical Induction

Long Answer Type

21.

Using principle of mathematical induction, prove that

$cos space straight alpha space space cos space 2 straight alpha space space cos space 4 straight alpha space............... cos left parenthesis 2 to the power of straight n minus 1 end exponent straight alpha right parenthesis space equals space fraction numerator sin space 2 to the power of straight n straight alpha over denominator 2 to the power of straight n sinα end fraction$ for all $straight n space element of space straight N$

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Short Answer Type

22.

Using principle of mathematical induction, show that $straight n less than 2 to the power of straight n space space for space all space straight n space element of space straight N$

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23.
Using principle of mathematical induction, show that $3 to the power of straight n greater than 2 to the power of straight n$ for all $straight n space element of space straight N$

Let P (n): $3 to the power of straight n greater than 2 to the power of straight n$

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

∴ P(1) is true

II. Suppose the statement is true for n = m, $straight m space element of space straight N$

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

III. For n = m + 1,

 $straight P left parenthesis straight m space plus space 1 right parenthesis space colon space space 3 to the power of straight m plus 1 end exponent space greater than space 2 to the power of straight m plus 1 end exponent$

 From (i), $3 to the power of straight m greater than 2 to the power of straight m$

$rightwards double arrow$ $3 to the power of straight m. space 3 greater than space 2 to the power of straight m.3 rightwards double arrow space 3 to the power of straight m plus 1 end exponent greater than 2 to the power of straight m. space left parenthesis 2 plus 1 right parenthesis space rightwards double arrow space 3 to the power of straight m plus 1 end exponent space greater than space 2 to the power of straight m plus 1 end exponent plus 2 to the power of straight m$

 Also, $2 to the power of straight m greater than 0 space rightwards double arrow space 3 to the power of straight m plus 1 end exponent space greater than space 2 to the power of straight m plus 1 end exponent$

∴ P (m + 1) is true.

∴ P(m) is true $rightwards double arrow$ P (m + 1) is true.

Hence, statement $straight P left parenthesis straight n right parenthesis space colon space 3 to the power of straight n greater than 2 to the power of straight n$ is true for all $straight n space element of space straight N.$

152 Views

Multiple Choice Questions

24.

If the number of terms in the expansion of $open parentheses 1 minus 2 over straight x space plus 4 over straight x squared close parentheses to the power of straight n comma straight x not equal to 0 comma$ is 28, then the sum of the coefficients of all the terms in this expansion is

64
2187
243
243

392 Views

25.

Statement − 1: For every natural number n ≥ 2 $fraction numerator 1 over denominator square root of 1 end fraction space plus space fraction numerator 1 over denominator square root of 2 end fraction space plus space..... space plus space fraction numerator 1 over denominator square root of straight n end fraction space greater than space square root of straight n$

Statement −2: For every natural number n ≥ 2, $straight n greater or equal than 2 comma space square root of straight n left parenthesis straight n plus 1 right parenthesis space end root space less than space straight n plus 1$

Statement −1 is false, Statement −2 is true
Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

125 Views

26.

If A = $open square brackets table row 1 0 row 1 1 end table close square brackets$ and I = $open square brackets table row 1 0 row 0 1 end table close square brackets$ , then which one of the following holds for all n ≥ 1, by the principle of mathematical induction

Aⁿ = n^A – (n – 1)I
Aⁿ = 2^n-1A – (n – 1)I
Aⁿ = nA + (n – 1)I
Aⁿ = nA + (n – 1)I

131 Views

27.

Let S(K) = 1 +3+5+..... (2K-1) = 3+K². Then which of the following is true?

S(1) is correct
Principle of mathematical induction can be used to prove the formula
S(K) ≠S(K+1)
S(K) ≠S(K+1)

158 Views

28.

Maximum sum of coefficient in the expansion of (1 – x sinθ + x² )ⁿ is

1
2ⁿ
3ⁿ
3ⁿ

126 Views

29.

For positive integer n, n³ + 2n is always divisible by

30.

The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is 8 m/s². The time taken by the particle to move the second metre is