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Ask iit jee aieee pet cbse icse state board community Community Discussion Question: progression 1
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dixit_se (0)

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prove that in any arithmetic progression, whose common difference is not equal to zero, the product of two terms equidistant from the extreme terms is greater the closer these terms are to the middle term
    
metal (491)

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Very easy!!!!!!!!
 
Take the first term to be a and the last term to be b. Let the common difference be  d .
then, mulitiplying 1st and last you get: ab
and multiplying 2nd and 2nd last you get ab+d(b-a-d).
Now, if d>0, b>a and also b> a+d
so, b-a-d >0
and hence d(b-a-d) > 0
If d<0
b<a
and b<a+d,
so, b-a-d<0
so d(b-a-d)>0
so, product of 2nd and 2nd last = ab+some positive number and is hence greater than ab[product of 1st and last]
So,
ab < ab+d(b-a-d)
so, ab+d(b-a-d)< ab + 2d(b-a-d)
i.e. product of 2nd and 2nd last< product of 3rd and 3rd last.
IN this way the product of equidistant terms keep on incresing as one moves closer to the middle term.
 
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