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.


< Previous Question

Next Question >

Study Tips for Answering NCERT Questions:

NCERT questions are designed to test your understanding of the concepts and theories discussed in the chapter. Here are some tips to help you answer NCERT questions effectively:

  • Read the question carefully and focus on the core concept being asked.
  • 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.
  • Practice answering questions within the time limit to improve your speed and accuracy.
  • Discuss your answers with your teachers or peers to get feedback and improve your understanding.

Comments

Comment(s) on this Question

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....

Chapters for Class 11 Mathematics

Subjects for Class 11


Notice: Undefined index: pageType in /var/www/html/ss_beta/saralstudy/ss_new/web/template/common/class_sidebar.php on line 45

Students Also Read

Important links

Quick Links

Resources

Support