The ratio of the sums of the first m and first n terms of an A.

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353.

The ratio of the sums of the first m and first n terms of an A. P. is m2: n2. Show that the ratio of its mth and nth terms is (2m−1):(2n−1). 


Let Sm and Sn be the sum of the first m and first n terms of the A.P. respectively. Let, a be the first terms and d be a common difference.

straight S subscript straight m over straight S subscript straight n space equals straight m squared over straight n squared
rightwards double arrow space fraction numerator begin display style straight m over 2 open square brackets 2 straight a plus left parenthesis straight m minus 1 right parenthesis straight d close square brackets end style over denominator begin display style straight n over 2 open square brackets 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d close square brackets end style end fraction space equals space straight m squared over straight n squared
rightwards double arrow space fraction numerator 2 straight a space plus left parenthesis straight m minus 1 right parenthesis straight d over denominator 2 straight a left parenthesis straight n minus 1 right parenthesis straight d end fraction space equals space straight m over straight n

⇒[2a+(m−1)d]
n =[2a(n−1)d]m
⇒2an+mnd−nd=2am+mnd−md
⇒md−nd=2am−2an
⇒(m−n)d=2a(m−n)
⇒d=2a
Now, the ratio of its mth and nth terms is
straight a subscript straight m over straight a subscript straight n space equals space fraction numerator straight a space plus left parenthesis straight m minus 1 right parenthesis straight d over denominator straight a plus left parenthesis straight n minus 1 right parenthesis straight d end fraction space equals space space fraction numerator begin display style straight a space plus left parenthesis straight m minus 1 right parenthesis 2 straight a end style over denominator begin display style straight a plus left parenthesis straight n minus 1 right parenthesis 2 straight a end style end fraction
space equals space fraction numerator straight a space left parenthesis 1 plus 2 straight m minus 2 right parenthesis over denominator straight a left parenthesis 1 plus 2 straight n minus 2 right parenthesis end fraction
space equals space fraction numerator 2 straight m minus 1 over denominator 2 straight n minus 1 end fraction

Thus, the ratio of its mth and nth terms is 2m – 1 : 2n – 1.

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