Question

the ratio of the sums of m and n terms of an A.P is mth and nth terms is (2m-1):(2n-1).​

Answer 1

Answer:

Sum of m terms of an A.P. = m/2 [2a + (m -1)d]

Sum of n terms of an A.P. = n/2 [2a + (n -1)d] m/2

[2a + (m -1)d] / n/2 [2a + (n -1)d] = m2 : n2

⇒ [2a + md - d] / [2a + nd - d] = m/n

⇒ 2an + mnd - nd = 2am + mnd - md

⇒ 2an - 2am = nd - md

⇒ 2a (n -m) = d(n - m)

⇒ 2a = d Ratio of m th term to n th term:

[a + (m - 1)d] / [a + (n - 1)d] =

[a + (m - 1)2a] / [a + (n - 1)2a] =

a [1 + 2m - 2] / a[1 + 2n -2] =

(2m - 1) / (2n -1)

So, the ratio of m th term and the n th term of the arithmetic series is (2m - 1) : (2n