C2 January 2006 Q4

EdexcelOld spec11 marksSequences & Series

4. The first term of a geometric series is 120. The sum to infinity of the series is 480.

(a) Show that the common ratio, \(r\), is \(\dfrac{3}{4}\). (3)

(b) Find, to 2 decimal places, the difference between the 5th and 6th term. (2)

The sum of the first \(n\) terms of the series is greater than 300.

(d) Calculate the smallest possible value of \(n\). (4)

Mark scheme (a)

Scheme	Marks
\(\dfrac{a}{1 - r} = 480\)	M1
\(\dfrac{120}{1 - r} = 480 \Rightarrow 120 = 480(1 - r)\)	M1
\(1 - r = \tfrac{1}{4} \Rightarrow \quad \underline{r = \tfrac{3}{4}} \quad *\)	A1cso
	(3)

Notes

1st M1 for use of \(S_\infty\)

2nd M1 substituting for \(a\) and moving \((1 - r)\) to form linear equation in \(r\).

For Information
\(u_1 = 120,\ u_2 = 90,\ u_3 = 67.5,\ u_4 = 50.625\)
\(S_2 = 210,\ S_3 = 277.5,\ S_4 = 328.125,\ S_5 = 366.09\ldots\)

Mark scheme (b)

Scheme	Marks
\(u_5 = 120 \times \left(\tfrac{3}{4}\right)^4\ [= 37.96875]\) \(u_6 = 120 \times \left(\tfrac{3}{4}\right)^5\ [= 28.4765625]\) either	M1
Difference \(= \underline{9.49}\) (allow \(\pm\))	A1
	(2)

Notes

M1 for some correct use of \(ar^{n-1}\). [\(120\left(\tfrac{3}{4}\right)^5 - 120\left(\tfrac{3}{4}\right)^6\) is M0]

Mark scheme (c)

Scheme	Marks
\(S_7 = \dfrac{120\left(1 - (0.75)^7\right)}{1 - 0.75}\)	M1
\(= 415.9277\ldots\) (AWRT) \(\underline{416}\)	A1
	(2)

Notes

M1 for a correct expression (need use of \(a\) and \(r\))

Mark scheme (d)

Scheme	Marks
\(\dfrac{120\left(1 - (0.75)^n\right)}{1 - 0.75} \gt 300\)	M1
\(1 - (0.75)^n \gt \dfrac{300}{480}\) (or better)	A1
\(n \gt \dfrac{\log(0.375)}{\log(0.75)}\) \((= 3.409\ldots)\)	M1
\(\underline{n = 4}\)	A1cso
	(4)
	(11 marks)

Notes

1st M1 for attempting \(S_n \gt 300\) [or \(= 300\)] (need use of \(a\) and some use of \(r\))

2nd M1 for valid attempt to solve \(r^n = p\ (r, p \lt 1)\), must give linear eqn in \(n\). Any correct log form will do.

Trial & Imp.

1st M1 for attempting at least 2 values of \(S_n\), one \(n \lt 4\) and one \(n \geqslant 4\).
2nd M1 for attempting \(S_3\) and \(S_4\).
1st A1 for both values correct to 2 s.f. or better.
2nd A1 for \(n = 4\).

For Information
\(S_2 = 210,\ S_3 = 277.5,\ S_4 = 328.125,\ S_5 = 366.09\ldots\)