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CBSE Class 10 Answered

If the pth term of an AP is 1/q and qth term 1/P. Prove that the sum of the first pqth term is 1/2 (pq + 1).
Asked by latayadav23 | 17 Oct, 2023, 08:39: PM
answered-by-expert Expert Answer
Let a be the first term and d be the common difference of Arithmetic progression
 
pth term  :-  a + (p-1)d = (1/q) ........................ (1)
 
qth term :-   a + ( q-1) d = ( 1/p ) .................... (2)
 
By subtracting eqn.(2) from eqn.(1) , we get
 
begin mathsize 14px style left parenthesis space p space minus space q space right parenthesis space d space equals space 1 over q minus fraction numerator 1 over denominator p space end fraction equals space fraction numerator p minus q over denominator p q end fraction end style
begin mathsize 14px style d space equals space fraction numerator 1 over denominator p space q end fraction end style
 
By substituting d in eqn.(1) , we get
 
begin mathsize 14px style a space plus space fraction numerator left parenthesis p minus 1 right parenthesis over denominator p space q end fraction space equals space 1 over q end style
 
from above expression , we get a = 1 / (pq)
 
Hence  sum S of (pq) terms is
 
begin mathsize 14px style S space equals space fraction numerator p q over denominator 2 end fraction open square brackets fraction numerator 1 over denominator p q end fraction plus space left parenthesis p q minus 1 right parenthesis fraction numerator 1 over denominator p q end fraction close square brackets space equals space 1 half left parenthesis p q plus 1 right parenthesis end style
 
Answered by Thiyagarajan K | 17 Oct, 2023, 11:03: PM

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