If a₁, a₂, a₃, a₄, a₅ are in H.P then a₁a₂ + a₂a₃ + a₃a₄ + a₄a₅ is equal to

$\begin{array}{l}\frac{1}{{\mathrm{a}}_2}-\frac{1}{{\mathrm{a}}_1}=\frac{1}{{\mathrm{a}}_3}-\frac{1}{{\mathrm{a}}_2}=\frac{1}{{\mathrm{a}}_4}-\frac{1}{{\mathrm{a}}_3}=\frac{1}{{\mathrm{a}}_5}-\frac{1}{{\mathrm{a}}_4}=\mathrm{d}\\ {\mathrm{a}}_1{\mathrm{a}}_2+{\mathrm{a}}_2{\mathrm{a}}_3+{\mathrm{a}}_3{\mathrm{a}}_4+{\mathrm{a}}_4{\mathrm{a}}_5=\frac{{\mathrm{a}}_1-{\mathrm{a}}_5}{\mathrm{d}}\\ and\frac{1}{{\mathrm{a}}_5}=\frac{1}{{\mathrm{a}}_1}+4\mathrm{d}\\ \Rightarrow {\mathrm{a}}_1-{\mathrm{a}}_5=4{\mathrm{da}}_1{\mathrm{a}}_5\Rightarrow \frac{{\mathrm{a}}_1-{\mathrm{a}}_5}{\mathrm{d}}=4{\mathrm{a}}_1{\mathrm{a}}_5\end{array}$