if there are 2n1terms in an ap prove that the

Question

If there are (2n+1)terms  in an AP ,prove that the ratio of the sum of odd terms and the sum of even terms is (n+1):n

Ans:    Let a, d be the I term & Cd of the AP.

∴ ak = a + (k - 1) d

s₁ = sum to odd terms

s₁ = a₁ +  a₃ + ......... a _{2n + 1}

s₁  = n +1/2 [a₁ + a_{2 n +1} ]

= n + 1/2 [2a₁ + 2nd]

s₁ = (n + 1) (a + nd)

s₂ = sum to even terms

s₂ = a₂ + a₄ + ..... a _2n

s₂  = n/2 [a₂ + a_2n  ]

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

=n [a + nd]

∴ s1 : s2 = (n +1)(a + nd )/n(a + nd )

= n+1/n