Theory question 109
Sequences and Series(a) The sum of the first n terms of a sequence is given by Sn = 5n2/2 + 5n/2. Write down the first four terms of the sequence and find an expression for the nth term.
(b) The equation of a circle is given by x2 + y2 − 10x − 8y + 25 = 0. (i) Show that the circle touches the X-axis. (ii) Find the coordinates of the point of contact.
Show the model answer
Model answer
(a) S1 = 5, S2 = 15, S3 = 30, S4 = 50
Terms: 5, 10, 15, 20
Tn = Sn − Sn−1 = 5n
(b)(i) Centre (5, 4), radius √(25 + 16 − 25) = 4; distance from centre to the x-axis = 4 = radius, so the circle touches it (or: y = 0 gives (x − 5)2 = 0, equal roots)
(ii) Point of contact (5, 0)
Explanation
Each term is the difference of two consecutive sums; a circle touches a line when the line is exactly one radius from the centre.