if the mth term of an ap is 1/n and nth term is 1/m, show that the sum of the mn terms is mn+1/2

Asked by afreensyed13 | 10th Feb, 2017, 06:11: PM

Expert Answer:

begin mathsize 16px style Given space that space straight m to the power of th space term equals 1 over straight n space and space straight n to the power of th space term equals 1 over straight m
To space prove colon Sum space of space mn space terms space of space the space AP equals space straight S subscript mn equals fraction numerator mn plus 1 over denominator 2 end fraction
Since space straight m to the power of th space term equals 1 over straight n space rightwards double arrow straight a plus left parenthesis straight m minus 1 right parenthesis straight d equals 1 over straight n rightwards double arrow an plus mnd minus nd equals 1 space.... left parenthesis straight i right parenthesis
and space straight n to the power of th space term equals 1 over straight m rightwards double arrow straight a plus left parenthesis straight n minus 1 right parenthesis straight d equals 1 over straight m rightwards double arrow am plus mnd minus md equals 1.... left parenthesis ii right parenthesis
rightwards double arrow an plus mnd minus nd equals am plus mnd minus md
rightwards double arrow an minus am equals nd minus md
rightwards double arrow straight a left parenthesis straight n minus straight m right parenthesis equals left parenthesis straight n minus straight m right parenthesis straight d
rightwards double arrow straight a equals straight d space... left parenthesis Since space straight n not equal to straight m right parenthesis
In space left parenthesis straight i right parenthesis comma space an plus mnd minus nd equals 1 space
rightwards double arrow nd plus mnd minus nd equals 1
rightwards double arrow mnd equals 1
rightwards double arrow straight d equals 1 over mn
Since space straight a equals straight d comma space so comma space straight a equals 1 over mn
straight S subscript mn equals mn over 2 left square bracket 2 straight a plus left parenthesis mn minus 1 right parenthesis straight d right square bracket
rightwards double arrow straight S subscript mn equals mn over 2 open square brackets 2 over mn plus fraction numerator left parenthesis mn minus 1 right parenthesis over denominator mn end fraction close square brackets space..... left parenthesis SInce space straight d equals straight a equals 1 over mn right parenthesis
rightwards double arrow straight S subscript mn equals fraction numerator mn over denominator 2 mn end fraction open square brackets 2 plus mn minus 1 close square brackets
rightwards double arrow straight S subscript mn equals 1 half open square brackets 1 plus mn close square brackets comma space that space is comma space straight S subscript mn equals fraction numerator mn plus 1 over denominator 2 end fraction
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Note:
Dear Student, next time please put proper brackets while writing your question. Sometimes it becomes difficult for us to understand. 
In the above problem, you wrote prove that the sum is mn+1/2, which means, (mn+1)/2.
The way you have written could also mean mn + 1/2(that is, mn+half). Just a suggestion so that we can help you as much as possible without any hindrance. Thank you.

Answered by Rebecca Fernandes | 10th Feb, 2017, 07:40: PM

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