(a) The sum of the first n terms of a sequence is given by (S_{n} = frac{5n^{2}}{2} + frac{5n}{2}). Write down the first four terms of the sequence and an expression for the nth term.
(b) The equation of a circle is given by (x^{2} + y^{2} – 10x – 8y + 25 = 0).
(i) Show that the circle touches the x- axis ; (ii) Find the coordinates of the point of contact.
Explanation
(a) (S_{n} = frac{5n^{2}}{2} + frac{5n}{2})
(S_{1} = frac{5(1^{2})}{2} + frac{5(1)}{2} = 5 implies T_{1} = 5)
(S_{2} = frac{5(2^{2})}{2} + frac{5(2)}{2} = 10 + 5 = 15)
(S_{3} = frac{5(3^{2})}{2} + frac{5(3)}{2} = 30)
(S_{4} = frac{5(4^{2})}{2} + frac{5(4)}{2} = 50)
(T_{2} = S_{2} – S_{1} = 15 – 5 = 10)
(T_{3} = S_{3} – S_{2} = 30 – 15 = 15)
(T_{4} = 50 – 30 = 20)
The first four terms are 5, 10, 15 and 20.
This is a linear sequence with d = 5.
(T_{n} = a + (n – 1)d)
= (5 + (n – 1)5 implies T_{n} = 5 + 5n – 5 = 5n)
(b) (x^{2} + y^{2} – 10x – 8y + 25 = 0)
For circle to touch x- axis, the distance from centre of circle to the x- axis must be equal to the radius.
(x^{2} + y^{2} – 10x – 8y + 25 = 0)
(2g = -10, g = -5)
(2f = -8 , f = -4)
Centre = (-g, -f) = (5, 4).
(Radius = sqrt{g^{2} + f^{2} – c} = sqrt{(-5)^{2} + (-4)^{2} – 25} = 4)
(|PQ| = sqrt{4^{2}} = 4)
Since PQ = radius, circle touches x- axis.