Question
[Maximum mark: 8]
Consider the geometric sequence 10, 5, 2.5, 1.25, …
(a) Express the general term nu in terms of n . [1]
(b) Find the first term which is smaller than 10-3 = 0.001. [3]
(c) Find the sum of the first 20 terms correct to 6 decimal places. [2]
(d) Find the sum of the infinite series.
Answer/Explanation
Answer:
(a) 10 x 0.5n-1 (= 20 x 0.5n )
(b) 0.000610
(c) 19.999981
(d) 20
Question
[Maximum mark: 6]
Find the sum of each of the following infinite geometric series
(i) \( 1+\frac{2}{5}+\frac{4}{25}+\frac{8}{125}+…\) (ii) \(1-\frac{2}{5}+\frac{4}{25}-\frac{8}{25}+…\)
Answer/Explanation
Answer:
(a) 5/3 (b) 5/7
Question
[Maximum mark: 6]
Calculate the following sums by using the appropriate formulas
(i) \(\sum_{k=1}^{6}4^{k}\) (ii) \(\sum_{k=11}^{6}(0.25)^{k}\) (correct to 6 dp) (iii) \(\sum_{k=11}^{+\infty }(0.25)^{k}\)
Answer/Explanation
Answer:
a. \(\frac{4(4^{6-1})}{4-1}=5460\) b. \(\frac{0.25(1-0.25^{6})}{1-0.25}=0.333252\) c. \(\frac{0.25}{1-0.25}=\frac{1}{3}\)
Question
Maximum mark: 5]
Consider the infinite geometric sequence 3, 3(0.9), 3(0.9)2, 3(0.9)3, … .
(a) Write down the 10th term of the sequence. Do not simplify your answer. [1]
(b) Find the sum of the infinite sequence. [4]
Answer/Explanation
Answer:
a. \(u^{10}=3(0.9)^{9}\)
b. \(S=\frac{3}{1-0.9}=\frac{3}{0.1}=30\)
Question
[Maximum mark: 6]
Consider the infinite geometric sequence 25, 5, 1, 0.2, … .
(a) Find the common ratio. [1]
(b) Find
(i) the 10th term; (ii) an expression for the nth term. [3]
(c) Find the sum of the infinite sequence. [2]
Answer/Explanation
Answer:
a. \(\frac{1}{5}(=0.2)\)
b. u10 \(=25\left ( \frac{1}{5} \right )^{9}=0.0000128\)
c. \(S=\frac{125}{4}=31.25\)