Question

यदि एक स० श्रेणी के प्रथम m तथा प्रथम n पदों के योगों का अनुपात m² :n² हो तो सिद्ध करें mवाँ पद : nवा पद =2m-1:2n-1​

Answer 1

Answer:

Step-by-step explanation:

Let first term  a and common difference d

given

$\frac{\frac{m}{2}[a+a_m] }{\frac{n}{2}[a+a_n]} = \frac{m^2}{n^2} \\\\\implies \frac{a+a_m}{a+a_n} =\frac{m}{n} \\\\\implies an+na_m= am +ma_n\\\\\implies an+n(a+(m-1)d)=am+m(a+(n-1)d)\\\implies 2a =d$

mth term will be  a+(m-1)d = a+ (m-1).2a = a ( 2m -1 )

simillary nth term = a (2n - 1 )

And their ratio

$\frac{a_m}{a_n}=\frac{2m-1}{2n-1} }$