Burner Problems
1.
!
The sum of the first 𝑛 terms of a geometric sequence is given by 𝑆! = Σ
% ' "
$ % .
"#$ & (
(a) Find the first term of the sequence, 𝑢$ .
[2]
(b) Find 𝑆) .
[3]
(c) Find the least value of 𝑛 such that 𝑆) − 𝑆! < 0.001.
[4]
2.
$
An infinite geometric series has first term 𝑢$ = 𝑎 and second term 𝑢% = * 𝑎% − 3𝑎, where 𝑎 > 0.
(a) Find the common ratio in terms of 𝑎.
[2]
(b) Find the values of 𝑎 for which the sum to infinity of the series exists.
[3]
(c) Find the value of 𝑎 when 𝑆) = 76.
[3]
3.
(b) The following diagram shows [CD], with length 𝑏 cm, where 𝑏 > 1. Squares with side lengths
𝑘 cm, 𝑘 % cm, 𝑘 & cm, …, where 0 < 𝑘 < 1, are drawn along [CD]. This process is carried on
indefinitely. The diagram shows the first three squares.
The total sum of the areas of all the squares is
+
$,
. Find the value of 𝑏.
[9]
4
5.
6.
7