Let p, q be integers and let $$\alpha $$, $$\beta $$ be the roots of the equation, x² $$-$$ x $$-$$ 1 = 0 where $$\alpha $$ $$ \ne $$ $$\beta $$. For n = 0, 1, 2, ........, let a_n = p$$\alpha $$ⁿ + q$$\beta $$ⁿ.

FACT : If a and b are rational numbers and a + b$$\sqrt 5 $$ = 0, then a = 0 = b.

a₁₂ = ?

Let p, q be integers and let $$\alpha $$, $$\beta $$ be the roots of the equation, x² $$-$$ x $$-$$ 1 = 0 where $$\alpha $$ $$ \ne $$ $$\beta $$. For n = 0, 1, 2, ........, let a_n = p$$\alpha $$ⁿ + q$$\beta $$ⁿ.

FACT : If a and b are rational numbers and a + b$$\sqrt 5 $$ = 0, then a = 0 = b.

If a₄ = 28, then p + 2q =

Consider an expanding sphere of instantaneous radius R whose total mass remains constant. The expansion is such that the instantaneous density $$\rho $$ remains uniform throughout the volume. The rate of fractional change in density $$\left( {{1 \over \rho } {{d\rho } \over {dt}}} \right)$$ is constant. The velocity $$v$$ of any point on the surface of the expanding sphere is proportional to

Consider regular polygons with number of sides $$n=3,4,5....$$ as shown in the figure. The center of mass of all the polygons is at height $$h$$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $$\Delta $$. Then $$\Delta $$ depends on $$n$$ and $$h$$ as