Permutations and Combinations

Permutations and Combinations

Question

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.

Answer

Here, a = 2, Let d be the common difference. Since the sum of first five terms is one-fourth of the sum of the next five terms.
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#6 {main}</pre>
or                       straight S subscript 5 space equals space 1 fourth left square bracket straight S subscript 10 space minus space straight S subscript 5 right square bracket            or              4 space straight S subscript 5 space equals space straight S subscript 10 space minus space straight S subscript 5        or         5 space straight S subscript 5 space equals space straight S subscript 10
or        5 5 over 2 left square bracket 2. space 2 space plus space left parenthesis 5 minus 1 right parenthesis straight d right square bracket space space equals space 10 over 2 left square bracket 2.2 space plus space left parenthesis 10 minus 1 right parenthesis straight d right square bracket              open curly brackets because space space straight S subscript straight n space equals space straight n over 2 left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket close curly brackets
or            space space 25 over 2 left square bracket 4 plus 4 straight d right square bracket space equals space 5 left square bracket 4 plus 9 straight d right square bracket          or                  50 + 50d = 20 + 45d
or                  5d = -30                             or                             d = -6
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#6 {main}</pre>                             open curly brackets because space space straight T subscript straight n space equals space straight a space plus space left parenthesis straight n minus 1 right parenthesis straight d close curly brackets
                           = 2 + 19 (-6) = 2 - 114 = -112

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