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Arithmetic Progressions

Question
CBSEENMA10007589

In an AP
Given d = 5, 59 = 75, find ‘a’ and ‘a’

Solution

Here,    d = 5, S= 75
We know that, 
   straight S subscript straight n equals straight n over 2 left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket
therefore space space space space straight S subscript 9 space equals space 9 over 2 space straight x space left square bracket 2 straight a plus left parenthesis 9 minus 1 right parenthesis space straight x space 5 right square bracket
rightwards double arrow space space space 75 space equals 9 over 2 straight x space left square bracket 2 straight a plus 8 space straight x space 5 right parenthesis
rightwards double arrow 150 space equals space 9 cross times left square bracket 2 straight a plus 40 right square bracket
rightwards double arrow 2 straight a plus 40 equals 150 over 9 equals 50 over 3
rightwards double arrow 2 straight a space equals space 50 over 3 minus 40 equals fraction numerator 50 minus 120 over denominator 3 end fraction
rightwards double arrow space straight a equals fraction numerator negative 70 over denominator 3 cross times 2 end fraction equals negative 35 over 3
We space know space that. space straight a subscript straight n equals straight a plus left parenthesis straight n minus 1 right parenthesis straight d
therefore space space space straight a subscript 9 equals negative 35 over 3 plus left parenthesis 9 minus 1 right parenthesis straight x 5
space space space space space space space space space equals negative 35 over 3 plus left parenthesis 9 minus 1 right parenthesis cross times 5
space space space space space space space space space space equals fraction numerator negative 35 over denominator 3 end fraction plus 40
space space space space space space space space space space space space space equals fraction numerator negative 35 plus 120 over denominator 3 end fraction equals 85 over 3
Hence comma space space space straight a equals fraction numerator negative 35 over denominator 3 end fraction space and space straight a subscript 9 equals 85 over 3