Permutations and Combinations

Permutations and Combinations

Question

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.

Answer

Here,              straight S subscript straight n space equals space An squared space plus space Bn                                                       ...(i)
Replacing n by (n - 1) in (i), we get
              straight S subscript straight n minus 1 end subscript space equals space straight A left parenthesis straight n minus 1 right parenthesis squared space plus space straight B left parenthesis straight n minus 1 right parenthesis space equals space straight A left parenthesis straight n squared minus 2 straight n plus 1 right parenthesis space plus space Bn space minus space straight B
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#6 {main}</pre>                                           ...(ii)
                  straight t subscript straight n space equals space straight S subscript straight n minus straight S subscript straight n minus 1 end subscript space equals space left parenthesis An squared plus Bn right parenthesis space minus space left parenthesis An squared minus 2 An plus straight A plus Bn minus straight B right parenthesis
                       = An squared plus Bn minus An squared plus 2 An minus straight A minus Bn plus straight B
                  straight t subscript straight n space equals space 2 An space minus space straight A space plus space straight B
Replacing n by (n - 1) in (iii), we get
                       straight t subscript straight n minus 1 end subscript space equals space 2 straight A left parenthesis straight n minus 1 right parenthesis space minus space straight A space plus space straight B space equals space 2 space An space minus space 2 straight A space minus space straight A space plus space straight B
Now,                    space space space straight d equals space straight t subscript straight n minus straight t subscript straight n minus 1 end subscript space equals space left parenthesis 2 space An space minus space straight A space plus space straight B right parenthesis space minus space left parenthesis 2 thin space An space minus space 3 straight A space plus space straight B right parenthesis
                                equals space 2 space An space minus space straight A space plus space straight B space minus space 2 space An space plus space 3 straight A space minus space straight B space equals space 2 straight A
which is independent of n

∴       The sequence is an A.P.
Yes, the converse is true.
Let a be the first term and d be the common difference.

∴              straight S subscript straight n space equals space straight n over 2 left square bracket 2 straight a space plus space left parenthesis straight n space minus space 1 right parenthesis straight d right square bracket space equals space an space plus space fraction numerator straight n squared straight d over denominator 2 end fraction space minus space straight n over 2 straight d space equals space open parentheses straight d over 2 close parentheses straight n squared space plus space open parentheses straight a minus straight d over 2 close parentheses straight n
                    equals space An squared space plus space Bn
where straight A space equals space straight d over 2 and straight B space equals space straight a space minus space straight d over 2 are fixed numbers.
Hence, if the sequence is an A.P. then the sum of first n terms must be of the form <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)
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#6 {main}</pre>




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