Page 47 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series
16. The r term, u , of an infinite series is given by:
th
r
1
r 1 2
u = 1 4r – 1 + 1 2 4r + 1
2
2
k
(a) Express u in the form 2 4r + 1 ,where k is a constant .
r
(b) Find the sum of the first n terms of the series, and deduce the sum of the infinite series.
17. (a) Find the smallest value of n such that the sum of the first n terms of the geometric progression
3
2
1 + 0.99 + (0.99) + (0.99) + …
is more than 75% of the sum to infinity.
th
(b) Find the n term and the sum of the first n terms of the series.
2 1 + 11 + 111 + 1111 + 11111 + …
18. By expressing the recurring decimals as the sum of a constant and an infinite geometric series, obtain
each of the following decimals as a fraction in its lowest terms.
· · ·
· ·
· ·
(a) 0.2731 (b) 0.597 5 (c) 0.7015 0 1
19. The sum of the first n terms of a sequence u , u , u , … is given by S = 3 [1 – 2 ].
–n
3
2
n
1
(a) Show that u = 3(2 –(n + 1) ). 2
n
(b) Find u n + 1 in terms of u , and deduce that the sequence is a geometric sequence.
n
(c) Determine the sum of the series u , u , u , ….
3
2
1
20. Show that 1 – 2 = 5x .
1 – 2x 2 + x (1 – 2x)(2 + x)
3
Hence, expand 5x in ascending powers of x up to the term in x .
(1 – 2x)(2 + x)
4
4
21. (a) Expand fully the expansions of ( 5 – 3) and ( 5 + 3) to evaluate ( 5 – 3) + ( 5 + 3) .
4
4
(b) Hence, using the inequality of 2 , 5 , 3 and the result from part (a), show that
751 , ( 5 + 3) , 752.
4
p
22. Write down the first four terms in ascending powers of x for the expansion of (1 + ax) . Given that the
first three terms of this expansion is 1 + 2x + 11 x , show that 2pa(pa – a) = 11 and find the value of a
2
4
and p. State the range of values of x for which the expansion is valid.
1
1
1
2
23. Show that 25 1 – 25 22 — can be expressed in the form n 39 , where n is an integer to be found.
1
—
2
2
Expand (1 – x) as a series in ascending powers of x up to the term in x . By using the first two terms
— 1 1 p
2
of the expansion of (1 – x) with x = 25 2 , obtain an approximate value for 39 in the form q , where

p and q are integers.
24. Two positive integers, p and q, are connected by p = q + 1. Using the binomial expansion, show that
the expression p – 2nq – 1 can be divided exactly by q for all positive integers n. By choosing suitable
2n
2
15
16
values for p and n, show that 3 – 33 can be divided exactly by 4, and hence, show that 3 + 5 can be
divided exactly by 4.
25. Use the binomial theorem to expand 1 + x as a series in ascending powers of x up to the term in x ,
2
1 – x
with |x| , 1. By substituting x = 1 into your result, show that 11 is approximately 663 .
10 200
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02 STPM Math T T1.indd 134 3/28/18 4:21 PM