Page 48 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

1
n
1
26. Show that the first three non-zero terms in the expansion of 1 – 1 2 — 1 in ascending powers of 1 2 is
n
n
1
1
1 2
1 – 1 2 2 – 1 1 3 , and find the term in 1 2 4 .

2 n
n
n
1
—
10
By giving n a suitable value, use the first four non zero terms of the series to find the value of (0.9) ,
giving your answer correct to five decimal places.
n 1 2n
2
2
27. (a) Given that ∑ r = n(n + 1)(2n + 1), obtain an expression, in its simplest terms, for ∑ (2r – 1) .
r = 1 6 r = n + 1
∞ 1
(b) Find ∑ = , expressing your answer as a fraction in its lowest terms. Hence, express the recurring
r = 1 10 3r
· · ·
decimal 0.1 0 8 as a fraction in its simplest form.
2
28. (a) Find the sum of the arithmetic progression
1, 4, 7, 10, 13, 16, …, 1000.
Now, each third term of the progression, i.e. 7, 16, …, is removed. Find the sum of the remaining
terms.
th
(b) The r term, u , of a series is given by
r
1
r 1 2
u = 1 3r – 2 + 1 2 3r – 1
n B 3 3
1
Express ∑ u in the form A 1 – n2 , where A and B are constants. Find A, B and the sum of the
r = 1 r 27
series.
1
—
2
4
29. (a) Find the binomial expansion of (1 + x) , for small values of x, up to the term in x , with coefficients
in their simplest forms. By substituting x = 1 in your expression, show that 17 ≈ 8317 .
4
16 4096
5

(b) Express (x + 2) – (x – 2) as a polynomial in x, and hence, find the exact value of ( 5 + 2) –
5
5
5
( 5 – 2) .
By assuming that 0 , 5 – 2 , 1 , deduce that the difference between ( 5 + 2) and an integer is
5
4
less than 1 .
1024
2
30. Express f(x) = x + 5x in the form A + B + C , where A, B and C are constants.
(1 + x)(1 – x) 2 1 + x 1 – x (1 – x) 2
If the expansion of f(x) in ascending powers of x is
r
2
3
c + c x + c x + c x + … + c x + … ,
r
2
0
3
1
find c , c , c and show that c = 11.
0
1
2
3
Express c in terms of r.
r
4
n
31. (a) Show that for a fixed number x ≠ 1, 2x + 2x + 2x + … + 2x is a geometric series, and find its sum in
3
2
terms of x and n.
(b) The series U (x) is given by
n
n
3
2
U (x) = x + 3x + 5x + … + (2n – 1)x , for x ≠ 1.
n
By considering U (x) – xU (x) and using the result from (a), show that
n
n
2
U (x) = x + x – (2n + 1)x n + 1 + (2n – 1)x n + 2
n
(1 – x)
2
15 14
r
Hence, determine the value of ∑ (2r – 1)3 and deduce the value of ∑ (2r + 1)3 r + 2 .
r = 1 r = 1
135
02 STPM Math T T1.indd 135 3/28/18 4:21 PM