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Let Tr be the rth term of an AP for r=1,2,3,.... If for some positive integers m, n. We have Tm=1/n and Tn=1/m, then Tmn equals to
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Asked by nitinsolanki14102001 | 16 Sep, 2023, 03:01: PM
answered-by-expert Expert Answer
 Let a be the initial value and d be the common difference of given arithmetic progression.
 
Let rth term of arithmetic progression Tr is
 
begin mathsize 14px style T subscript r space equals space a space plus space left parenthesis r minus 1 right parenthesis space d end style
Given mth and nth terms are
 
begin mathsize 14px style T subscript m space equals space a space plus space left parenthesis m minus 1 right parenthesis space d space equals space 1 over m end style
begin mathsize 14px style T subscript n space equals space a space plus space left parenthesis n minus 1 right parenthesis space d space equals space 1 over n end style
By solving above simultaneous linear equations in a and d, we get
 
begin mathsize 14px style a space equals space fraction numerator m plus n minus 1 over denominator m space n end fraction end style
begin mathsize 14px style d space equals negative space fraction numerator 1 over denominator m space n end fraction end style
 
Then we get Tmn as
 
begin mathsize 14px style T subscript m n end subscript space equals space a space plus space left parenthesis space m space n space minus space 1 space right parenthesis space d end style
 
begin mathsize 14px style T subscript m n end subscript space equals space fraction numerator m plus n minus 1 over denominator m space n end fraction space minus space left parenthesis space m space n space minus 1 space right parenthesis space fraction numerator 1 over denominator m n end fraction space end style
begin mathsize 14px style T subscript m n end subscript space equals space fraction numerator m space plus space n space over denominator m n end fraction space minus space 1 end style
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