Page 46 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

7. The arithmetic mean of 1 and 1 is 1 .
a + b b + c a + c
2
2
Find, in terms of b, the arithmetic mean of a and c .
th
8. Find the smallest value of n such that the n term of the arithmetic series 12.0 + 10.7 + 9.4 + 8.1 + …
is negative. Find, also, the smallest value of n such that the sum to n terms is negative.

2
9. The sum of the first n terms of an arithmetic sequence u , u , u , … is given by S = 2n + n. Find an
3
n
1
2
explicit formula and recursive formula for u .
n
10. A sequence is u , u , u , … is defined by u = 4n – 1. The difference between successive terms of the
2
n
3
1
2
sequence forms a new sequence w , w , w , …. 2
2
1
3
(a) Find an explicit formula for w in terms of n.
n
(b) Show that w , w , w , … forms an arithmetic sequence, and state its first term and common difference.
3
2
1
(c) Find the sum of the first n terms of the sequence w , w , w , … in terms of u and w .
n
n
1
2
3
11. If S represents the sum of the first n terms of a geometric progression
n
1
1
1 + 1 + 1 2 2 + … + 1 2 n – 1 + …
2 2 2
and S represents the sum to infinity, find the smallest value of n such that S – S , 0.001.
n
3
12. A geometric sequence is defined by U r + 1 = 2 + U , and U = 2.
r
1
5
r – 1
3
(a) Write down each of the terms U , U and U in the form ∑ 2 1 2 m , and show that explicit formula
3
5
4
2
m = 0
3
r
3
for U is given by U = 5 1 – 1 24 .
r
r
5
(b) Determine the limit of U when r tends to infinity.
r
2
n
13. If S is the sum of the series 1 + 3x + 5x + … + (2n + 1)x , by considering (1 – x)S, show that
S = 1 + x – (2n + 3)x n + 1 + (2n + 1)x n + 2 , x ≠ 1.
(1 – x) 2
14. By bracketing the terms into pairs, show that
2
2
1 – 2 + 3 – 4 + … + (2n – 1) – (2n) = –n(2n + 1).
2
2
2
2
Deduce the sum of the series
2
2
2
2
2
2
2
(a) 1 – 2 + 3 – 4 + … + (2n – 1) – (2n) + (2n + 1) ,
2
2
2
2
2
2
(b) 25 – 26 + 27 – 28 + … + 49 – 50 .
15. The series of positive integers is grouped into four as follows:
(1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), …
Show that the sum of the integers in the k bracket is 2(8k – 3).
th
If the integers are similarly grouped with m integers in each bracket, find
(a) the last term in the n bracket,
th
(b) the first term in the n bracket.
th
th
If S represents the sum of the integers in the n bracket, find an expression for S in terms of m and n.
n
n
Hence, show that S , S , S are in arithmetic progression.
3n
2n
n
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02 STPM Math T T1.indd 133 3/28/18 4:21 PM