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Sequences and Series

Short Answer Type

21.

The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r) a + (r – p) b + (p – q)c = 0

180 Views

22.

Show that the sum of (m + n)^th and (m – n)^th terms of an A.P. is equal to twice the m^th term.

157 Views

23.

Find the sum to n terms of an A.P., whose nth term is $straight t subscript straight n space equals space 5 space minus space 6 straight n comma space straight n element of space straight N.$

143 Views

24.
Solve the equation 1 + 6 + 11 + 16 + ........ + x = 148

The given equation is : 1 + 6 + 11 + 16 +...........+x = 148

Here, a = 1, d = 6 -1 = 5, l = x, Let x be the nth term

∴ $straight S subscript straight n space equals space 148$

or $straight n over 2 left square bracket 2 straight a space plus space left parenthesis straight n minus 1 right parenthesis straight d right square bracket space equals space 148$

or $straight n over 2 left square bracket 2 left parenthesis 1 right parenthesis space plus space left parenthesis straight n space minus space 1 right parenthesis 5 right square bracket space equals space 148$

or n[2 + 5n - 5] = 296

or n (5n - 3) = 296

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

or $straight n equals space fraction numerator 3 plus-or-minus square root of left parenthesis 3 right parenthesis squared minus 4 left parenthesis 5 right parenthesis space left parenthesis negative 296 right parenthesis end root over denominator 2.5 space end fraction space equals space fraction numerator 3 plus-or-minus square root of 9 plus 5920 end root over denominator 10 end fraction space equals space fraction numerator 3 plus-or-minus square root of 5929 over denominator 10 end fraction space equals space fraction numerator 3 plus-or-minus 77 over denominator 10 end fraction space equals space 8 comma space space fraction numerator negative 37 over denominator 5 end fraction$

$rightwards double arrow$ n = 8 (∵ n is a natural number)

∴ $straight t subscript 8 space equals space straight x$ or a + (8 -1)d = x or 1+7 (5) = x or x = 36

140 Views

25.

How many terms of the A.P. -6, $fraction numerator negative 11 over denominator 2 end fraction comma$ -5, .......... are needed to give the sum -25? Explain the double answer.

962 Views

26.

In a A.P., the first term is 2 and the sum of first five terms is one-fourth of the sum of the next five terms. Show that the 20th term is – 112.

133 Views

Long Answer Type

27.

If the sum of first n terms of a sequence is of the form An² + Bw, where A and B arc constants (independent of n). Show that the sequence is an A.P. Is the converse true? Justify your answer.

189 Views

Short Answer Type

28.

In an A.P., if the pth term is $1 over straight q$ and the qth term is $1 over straight p comma$ prove that the sum of the first pq terms must be $1 half left parenthesis pq space plus space 1 right parenthesis.$ where $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

129 Views

29.

If the pth term of an A.P. is a and qth term is b, show that the sum of (p + q) terms

is $fraction numerator straight p plus straight q over denominator 2 end fraction open square brackets straight a plus straight b plus fraction numerator straight a minus straight b over denominator straight p minus straight q end fraction close square brackets$