Question

If there are (2n+1) terms in A.P.,then prove that the ratio of the sum of odd terms and the sum of even terms is (n+1):n

Answer 1

for odd term,
a =1
d=2
we use formula ,
Sn=n/2 {2a+(n-1) d}

now ,
S=(2n+1)/2 {2 x 1 +(2n+1-1) 2}
=(2n+1)(1+2n)=(1+2n)^2----------------(1)

for even
a=2
d=2
S'=(2n+1)/2 {2 x 2 +(2n+1-1) 2 }
= (2n+1)(2+2n)=2 (1+n)(2n+1) ------------(2)
now sum of odd /sum of even=(2n+1)(2n+1)/(2n+1)(2n)
=(2n+1)/2n