Let the arithmetic mean of a1 and b1 be 165,a>2. If α is such that a,4,α, b are in A.P., then the equation αx2−ax+2(α−2b)=0 has:
Held on 28 Jan 2026 · Verified 6 Jul 2026.
one root in (0,2) and another in (−4,−2)
one root in (1,4) and another in (-2,0)
both roots in the interval (−2,0)
complex roots of magnitude less than 2
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Let $a_{1}=1$ and for $n \geqslant 1, a_{n+1}=\frac{1}{2} a_{n}+\frac{n^{2}-2 n-1}{n^{2}(n+1)^{2}}$. Then $\left|\sum_{n=1}^{\infty}\left(a_{n}-\frac{2}{n^{2}}\right)\right|$ is equal to $\_\_\_\_$.
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