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
s_{1} = sum to odd terms
s_{1} = a_{1} + a_{3} + ......... a_{ 2n + 1}
s_{1} = n +1/2 [a_{1} + a_{2 n +1 }]
= n + 1/2 [2a_{1} + 2nd]
s_{1} = (n + 1) (a + nd)
s_{2} = sum to even terms
s_{2} = a_{2} + a_{4} + ..... a _{2n}
s_{2} = n/2 [a_{2} + 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