Question

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

Solution

Here, $straight t subscript straight n space equals space 5 space minus space 6 straight n$ ...(i)
Replacing n by (n - 1), we get
$straight t subscript straight n minus 1 end subscript space equals space 5 space minus space 6 left parenthesis straight n space minus space 1 right parenthesis space equals space 5 space minus space 6 straight n space plus space 6 space equals space 11 space minus space 6 straight n$ ...(ii)

∴ d = $straight t subscript straight n space minus space straight t subscript straight n minus 1 end subscript space equals space left parenthesis 5 minus 6 straight n right parenthesis space minus space left parenthesis 11 space minus space 6 straight n right parenthesis space equals space 5 space minus space 6 straight n space minus space 11 space plus space 6 straight n space equals space minus 6$

which is independent of n

∴ The sequence $space space open curly brackets straight t subscript straight n close curly brackets$ is an A.P.
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 a = -1, d = -6
$straight S subscript straight n space equals space straight n over 2 left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket space equals space straight n over 2 left square bracket 2 left parenthesis negative 1 right parenthesis space plus space left parenthesis straight n minus 1 right parenthesis space left parenthesis negative 6 right parenthesis right square bracket space space space space space space space equals space straight n over 2 left square bracket negative 2 minus 6 straight n plus 6 right square bracket space equals space straight n over 2 left square bracket 4 minus 6 straight n right square bracket space equals space straight n left parenthesis 2 minus 3 straight n right parenthesis space equals space 2 straight n space minus space 3 straight n squared$

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Permutations And Combinations

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

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