The ratio of the sums of m and n terms of an A.P. is m:n?. Show that the value
of the m and n'th terms is (2m - 1): (2n-1).
Answers
Given : The ratio of the sums of m and n terms of an A.P. is m²:n² ( correction)
To find : Ration of value of the m and nth terms is (2m - 1): (2n-1)
Solution:
Sum of m terms = (m/2 )(2a + (m - 1)d)
Sum of m terms = (n/2 )(2a + (n - 1)d)
(m/2 )(2a + (m - 1)d )/ (n/2 )(2a + (n - 1)d) = m²/n²
=> (2a + (m - 1)d )/ (2a + (n - 1)d) = m/n
=> n (2a + (m - 1)d ) = m (2a + (n - 1)d)
=> 2an + nmd - nd = 2am + mnd - md
=> 2an - nd = 2am - md
=> d(m - n) = 2a(m - n)
=> 2a = d
mth term = a + (m - 1)d = a + (m - 1)2a =a(1 + 2m - 2) = a(2m - 1)
nth term = a + (n - 1)d = a + (n - 1)2a =a(1 + 2n - 2) = a(2n - 1)
mth term : nth term = a(2m - 1) : a(2n - 1)
=> mth term : nth term = (2m - 1) : (2n - 1)
QED
Learn more:
किसी समांतर श्रेणी के m तथा n पदों के योगफलों का ...
https://brainly.in/question/9240400
The Ratio of Sumof n terms of the twoAP's is (n+1) :(n-1)then the ...
https://brainly.in/question/12937646