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CBSE Class 10 Answered

if m times the m th term of an AP is equal to n times the n th term, show that the (m+n) th term of the AP is zero
Asked by prassanna.j | 13 Nov, 2023, 11:23: PM
answered-by-expert Expert Answer
Let a be the first term of Arithmetic Progression( AP )  and d be the common difference .
 
m-th term of AP is
 
begin mathsize 14px style t subscript m space equals space a space plus space left parenthesis space m minus 1 space right parenthesis space d end style
 
m times of m-th term of AP is
 
begin mathsize 14px style m space t subscript m space equals space m space a space plus space m space left parenthesis space m minus 1 space right parenthesis space d end style .........................(1)
 
n-th term of AP is
 
begin mathsize 14px style t subscript n space equals space a space plus space left parenthesis space n minus 1 space right parenthesis space d end style
 
m times of m-th term of AP is
 
begin mathsize 14px style n space t subscript n space equals space n space a space plus space n space left parenthesis space n minus 1 space right parenthesis space d end style .........................(2)
 
If  ( m tm ) = ( n tn ) then from eqn.(1) and (2) , we get
 
begin mathsize 14px style m space a space plus space m space left parenthesis m minus 1 right parenthesis space d space equals space n space a space plus space n space left parenthesis n minus 1 right parenthesis space d end style
begin mathsize 14px style left parenthesis m minus n right parenthesis space a space plus space left parenthesis space m squared minus n squared right parenthesis d space minus left parenthesis m minus n right parenthesis d space equals space 0 end style
Dividing by (m-n) , we get
 
begin mathsize 14px style a space plus space left parenthesis space m plus n space minus 1 space right parenthesis space d space equals space 0 end style
 
LHS of above expression is (m+n)th term of AP , i.e., t(m+n)
 
Hence (m+n)th term of AP is zero



Answered by Thiyagarajan K | 14 Nov, 2023, 11:07: AM

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