If $a, b, c$ are in A.P. and if the equations $(b-c) x^2+(c-a) x+(a-b)=0$ and $2(c+a) x^2+(b+c) x=0$ have a common root, then
Given an A.P. and a G.P. with positive terms, with the first and second terms of the progressions being equal. If $$a_n$$ and $$b_n$$ be the $$n^{\text {th }}$$ term of A.P. and G.P. respectively then
If for the series $$a_1, a_2, a_3$$, ...... etc, $$\mathrm{a}_{\mathrm{r}}-\mathrm{a}_{\mathrm{r}+\mathrm{i}}$$ bears a constant ratio with $$\mathrm{a}_{\mathrm{r}} \cdot \mathrm{a}_{\mathrm{r}+1}$$; then $$\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3 \ldots .$$. are in
If $$\alpha_1, \alpha_2, \ldots, \alpha_n$$ are in A.P. with common difference $$\theta$$, then the sum of the series $$ \sec \alpha_1 \sec \alpha_2+\sec \alpha_2 \sec \alpha_3+\ldots .+\sec \alpha_{n-1} \sec \alpha_n=k\left(\tan \alpha_n-\tan \alpha_1\right)$$ where $$\mathrm{k}=$$