Page 11 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

Exercise 2.1

th
1. In each of the sequences below, write down the next two terms and state the n term.
(a) 5, 10, 15, 20, … (b) 4, 7, 10, 13, … (c) 2, 5, 8, 11, …
3
(d) 2, 4, 8, 16, … (e) 3, – , 3 , – 3 , … (f) 1 , 3 , 5 , 7 , …
5 25 125 3 5 7 9
,
(g) 1, 5 , 25 125 , … (h) 1 , 1 , 1 , 1 , … (i) 1, –2, 3, –4, …

2 4 8 2 6 12 20
2. Write down the first four terms of each of the following sequences.
(a) 5 n 6 (b) 5 (–1) n + 1 6 (c) 5 (2x) n – 1 26
2 n + 1 n! (2n – 1)
n 2n – 1
(–1) x
n 6
(d) 5 (2n – 1)(2n + 1) 6 (e) 5 cos nx
nx
3. Find the n term of each of the following sequences.
th
1
(a) – , 3 , – 5 , 7 , – 9 , … (b) 1, 0, 1, 0, 1, … (c) 2 , 0, 3 , 0, 4 , …
5 8 11 14 17 3 4 5
4. Find the next five terms of each of the sequences below, defined by the recursive formula given.
n
(a) u n + 1 = nu – n, u = 3. (b) u n + 1 = 2u – (–1) , u = 1.
n
1
n
1
(c) u n + 2 = 3u n + 1 – 2u , u = 1, u = 3.
2
n
1
5. Determine if each of the following sequences are convergent or divergent. For those that are convergent,
find their limits.
(a) 1 + 1 , 1 + 1 , 1 + 1 , … (b) 1 + 1 , 2 + 1 , 3 + 1 , …
2 3 4 2 3 4
(c) 2 , 2 , 2 , … (d) 3 – 1 , 3 – 2 , 3 – 3 , …
1 + 1 1 + 1 1 + 1 2n 3n 4n
2 3 4
6. Evaluate
2
3
1
(a) lim 1 2 (b) lim 1 2 2 (c) lim n + 2 2
n → ∞ n n → ∞ n – 1 n → ∞ n
2
1
(d) lim n + 2 2 (e) lim 1 2n + 2n 2 (f) lim 1 1 + 2 × 5 n n 2
n → ∞ n – 1 n → ∞ 5n – n n → ∞ 4 + 3 × 5
2
7. Write each of the following series by using the ∑ notation.
(a) 1 + 8 + 27 + 64 + … + 1 000 (b) 1 + 3 + 5 + 7 + … + 99
(c) 1 + 1 + 1 + 1 + … + 1 (d) 1 – 1 + 1 – 1 + 1 – 1
2 4 8 512 3 9 27 81 243
(e) 11 + 8 + 5 + 2 – 1 – 4 – 7 (f) 5 + 7 + 11 + 19 + 35 + 67
3
2
(g) –x + 2x – 3x + … + 10x 10 (h) 1 + 2 + 3 + … + 8
2 · 3 3 · 4 4 · 5 9 · 10
8. Write down the terms of each of the following series without the ∑ notation.
10 8 1 5
(a) ∑ (r – r) (b) ∑ (c) ∑ (3r + 2)
2
r = 1 r = 1 r 2 r = 1
9 10 7
(d) ∑ (–1) r – 1 (3r) (e) ∑ (8 – r) (f) ∑ (r + 2)(r + 4)
r = 1 r = 1 r = 1
5 5
2
(g) ∑ (6 – r) 2 (h) ∑ (2r – 3r – 5)
r = 1 r = 1
98

02 STPM Math T T1.indd 98 3/28/18 4:21 PM