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Sum of the first p, q and r terms of an A.P. are&n...

Sum of the first p, q and | Class 11 Mathematics Chapter Sequence and Series, Sequence and Series NCERT Solutions

Question:

Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.

Prove that $a over p open parentheses q minus r close parentheses plus b over q open parentheses r minus p close parentheses plus c over r open parentheses p minus q close parentheses equals 0$

Answer:

Let a₁ and d be the first term and the common difference of the A.P. respectively.

According to the given information,

$S subscript p space end subscript equals space p over 2 open square brackets 2 a subscript 1 plus space open parentheses p minus 1 close parentheses d close square brackets equals a rightwards double arrow 2 a subscript 1 space plus space open parentheses p minus 1 close parentheses d space equals space fraction numerator 2 a over denominator p end fraction space space space space space space... left parenthesis 1 right parenthesis S subscript q equals space q over 2 open square brackets 2 a subscript 1 plus space open parentheses q minus 1 close parentheses d close square brackets equals b rightwards double arrow 2 a subscript 1 space plus space open parentheses q minus 1 close parentheses d space equals space fraction numerator 2 b over denominator q end fraction space space space space space space... left parenthesis 2 right parenthesis S subscript r equals space r over 2 open square brackets 2 a subscript 1 plus space open parentheses r minus 1 close parentheses d close square brackets equals c rightwards double arrow 2 a subscript 1 space plus space open parentheses r minus 1 close parentheses d space equals space fraction numerator 2 c over denominator r end fraction space space space space space space... left parenthesis 3 right parenthesis S u b t r a c t i n g space left parenthesis 2 right parenthesis space f r o m space left parenthesis 1 right parenthesis comma space w e space o b t a i n left parenthesis p minus 1 right parenthesis d space minus left parenthesis q minus 1 right parenthesis d equals fraction numerator 2 a over denominator p end fraction equals fraction numerator 2 b over denominator q end fraction rightwards double arrow d open parentheses p minus 1 minus q plus 1 close parentheses equals fraction numerator 2 a q minus 2 b q over denominator p q end fraction rightwards double arrow d left parenthesis p minus q right parenthesis equals fraction numerator 2 a q minus 2 b p over denominator p q end fraction rightwards double arrow d equals fraction numerator 2 open parentheses a q minus b p close parentheses over denominator p q open parentheses p minus q close parentheses end fraction space space space space space space space space space space space... left parenthesis 4 right parenthesis S u b t r a c t i n g space left parenthesis 3 right parenthesis space f r o m space left parenthesis 2 right parenthesis comma space w e space o b t a i n open parentheses q minus 1 close parentheses d minus open parentheses r minus 1 close parentheses d equals fraction numerator 2 b over denominator q end fraction minus fraction numerator 2 c over denominator r end fraction rightwards double arrow d open parentheses q minus 1 minus r plus 1 close parentheses equals fraction numerator 2 b over denominator q end fraction minus fraction numerator 2 c over denominator r end fraction rightwards double arrow d open parentheses q minus r close parentheses equals fraction numerator 2 b r minus 2 q c over denominator q r end fraction rightwards double arrow d equals fraction numerator 2 open parentheses b r minus q c close parentheses over denominator q r open parentheses q minus r close parentheses end fraction space space space space space space space space space space space... left parenthesis 5 right parenthesis E q u a t i n g space b o t h space t h e space v a l u e space o f space d space o b t a i n e d space i n space left parenthesis 4 right parenthesis space a n d space left parenthesis 5 right parenthesis comma space w e space o b t a i n fraction numerator a q minus b p over denominator p q open parentheses p minus q close parentheses end fraction equals fraction numerator b r minus q c over denominator q r open parentheses q minus r close parentheses end fraction rightwards double arrow q r open parentheses q minus r close parentheses open parentheses a q minus b q close parentheses equals p q open parentheses p minus q close parentheses open parentheses b r minus q c close parentheses rightwards double arrow r open parentheses a q minus b p close parentheses open parentheses q minus r close parentheses equals p open parentheses b r minus q c close parentheses open parentheses p minus q close parentheses rightwards double arrow open parentheses a q r minus b p r close parentheses open parentheses q minus r close parentheses equals open parentheses b p r minus p q c close parentheses open parentheses p minus q close parentheses D i v i d i n g space b o t h space s i d e s space b space p q r ; w e space o b a t i n open parentheses a over p minus b over q close parentheses open parentheses q minus r close parentheses equals open parentheses b over q minus c over r close parentheses open parentheses p minus q close parentheses rightwards double arrow a over p open parentheses q minus r close parentheses minus b over q open parentheses q minus r plus p minus q close parentheses plus c over r open parentheses p minus q close parentheses equals 0 rightwards double arrow a over p open parentheses q minus r close parentheses plus b over q open parentheses r minus p close parentheses plus c over r open parentheses p minus q close parentheses equals 0 T h u s comma t h e space g i v e n space r e s u l t space i s space p r o v e d.$

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Reference examples and data from the chapter when answering questions about Sequence and Series.
Review previous year question papers to get an idea of how such questions may be framed in exams.
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Welcome to the NCERT Solutions for Class 11 Mathematics - Chapter . This page offers a step-by-step solution to the specific question from Excercise 2 , Question 11: Sum of the first p, q and r terms of an A.P. are a, b and c, resp....

Sum of the first p, q and | Class 11 Mathematics Chapter Sequence and Series, Sequence and Series NCERT Solutions

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