If three distinct numbers a,b and c are in GP and the equation ax2+2bx+c=0 and dx2+2ex+f=0 have a common root, then which of the following statements is correct?
d,e,f are in A.P
\(\frac{d}{a},\frac{e}{b},\frac{f}{c}\) are in A.P
d,e,f are in G.P
\(\frac{d}{a},\frac{e}{b},\frac{f}{c}\) are in G.P
a,b, and c are three distinct numbers and from the equation –

If the sum of the first 10 terms of the series \[ \frac{4 \cdot 1}{1 + 4 \cdot 1^4} + \frac{4 \cdot 2}{1 + 4 \cdot 2^4} + \frac{4 \cdot 3}{1 + 4 \cdot 3^4} + \ldots \] is \(\frac{m}{n}\), where \(\gcd(m, n) = 1\), then \(m + n\) is equal to _____.
If \(\sum\)\(_{r=1}^n T_r\) = \(\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\) , then \( \lim_{n \to \infty} \sum_{r=1}^n \frac{1}{T_r} \) is equal to :