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In an A.P. with first term 'a' and the c...

In an `A.P.` with first term 'a' and the common difference d `(a, d!=0)`, the ratio `'rho`' of the sum of the first n terms to sum of n terms succeeding them does not depend on n. Then the ratio `a/d` and the ratio, `'rho'` ,respectively are

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`Sn=n/2(2a+(n-1)d)`
`S2n= 2n/2(2a+(2n-1)d)`
`rho=(Sn)/(S2n-Sn)`
`rho= (((n/2)(2a+(n-1)d))/(n(a+(3n/2-1/2)d)))`(it should be independent of n) Therefore, 2a-d=0
a/d=1/2
rho= (1/2*(dn))/(3nd/2)=1/3
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