Page 32 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series
6. Find the sum of each of the following series.
n 2n
(a) ∑ (3r – 1) 2 (b) ∑ r(r + 2)(r + 5)
r = 1 r = 1
2n 3n
2
(c) ∑ r(2r + 3) (d) ∑ r(r + 1)
r = n r = 1
n n
2
3
3
7. Using the identity r – (r – 1) ≡ 4r – 6r + 4r – 1 and ∑ r = 1 n(n + 1)(2n + 1), find ∑ r .
4
4
2
r = 1 6 r = 1
th
8. Using the method of differences, find the sum of the first n terms of the series whose r term, u , are as
r
follows.
2
1
(a) u = (r + 1)(r + 2) (b) u = (r + 2)(r + 3) 2
r
r
2r – 1
1
(c) u = r(r + 3) (d) u = r(r + 1)(r + 2)
r
r
9. Find the sum to n terms of the series.
2
2
2
2
(a) 1 · 5 + 5 · 9 + 9 · 13 + 13 · 17 + …
(b) 1 + 1 + 1 + 1 + …
1 · 3 · 4 2 · 4 · 5 3 · 5 · 6 4 · 6 · 7
n
10. If f(r) = 1 , simplify f(r) – f(r + 1). Hence, find ∑ r .
r! r = 1 (r + 1)!
n 2n
2
11. If ∑ u = 3n + 2n, find u . Hence, find ∑ u .
r = 1 r r r = n + 1 r
12. If f(r) = 1 , simplify f(r) – f(r + 1). Hence, find the sum of the first n terms of the series
r 2 3 5 7 …
2
2
1 · 2 2 + 2 · 3 2 + 3 · 4 2 +
2
13. Find the numbers A, B and C such that
1 + r ≡ A(r + 2)(r + 1) + B(r + 1) + C for all values of r.
2
Hence, prove that
n
∑ (1 + r ) r! = n(n + 1)!
2
r = 1
2.3 Binomial Expansions
n
The 1 2 and n! notation
r
Consider the product of the first 50 positive integers as follows:
50 × 49 × 48 × … × 3 × 2 × 1.
The product of these integers is a huge number. To simplify and for ease of writing, the product of these numbers
may be written as 50! (which is read as “factorial fifty”).
Hence,
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
= 3 628 800

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