A#38
paper 1 day 4
Topic 1 Algebra
Paper 1 Sequences
(use a scientific calculator)
(None in 2008 p1 TZ2, 2008 p2 TZ1, 2008 p2 TZ0)
Topic 1 Algebra
2005 p1 TZ0 #1
Let Sn be the sum of the first n terms of an arithmetic
sequence, whose first three terms are u1, u2 and u3. It is
known that S1 = 7 and S2 = 18.
a) Write down u1.
b) Calculate the common difference of the sequence.
c) Calculate u4.
Series and sequences
2005 p1 TZ0 #1
(a)
(c)
u1 = S1 = 7
(b) S2 = u1 + u2 = 18
= 7 + u2 = 18
u2 = 11
u1, u2, u3, u4, …
7, 11, 15, 19, …
d=4
The 4th term is
And the common
difference is:
d = u2 – u1
= 11 – 7
d=4
Arithmetic sequence
u4 = 19
2008 p1 TZ1 #3
Series and sequences
2008 p1 TZ1 #3
(a)
(b) un=152= u1 + (n – 1)d
u1 = 2
u2 = 5
d =3
n = 101
u1 = 2
152 = 2 + (n –
u2 1)
= 53
d =3
150 = (n – 1) 3
n =
d = u2 – u1
= 5 – 2
d= 3
50 = (n – 1)
un = u1 + (n – 1 )d
u101 = 2 + (101 – 1) 3
u101 = 302
n = 51
st term is 152:
The
u1,u51
2,…, u51, …
2, 5,…, 152, …
d=3
Arithmetic sequence
2007 p1 TZ1 #1
Series and sequences
2007 p1 TZ1 #1
(a)
(b) (i)
u1, u2, u3,u4, … u = 25 un = u1 rn – 1
1
u1 = 25
25, 5, 1, .2, … u = 5
2
u10 = 25 (0.2)9
u2 = 5
r = 0.2
r =
=
0.0000128
n = 10
= 1.28 x 10-5
The common ratio is
(b) (ii)
u2 5
r 
The nth term is
u1 25
n-1
u
=
25(0.2)
n
1
  0.2
5
Geometric sequence
Series and sequences
2007 p1 TZ1 #1
(c)
u1 = 25
u2 = 5
r = 0.2
u1, u2, u3,u4, …
25, 5, 1, .2, …
r=0.2
The infinite sum is
u1
S 
,| r | 1
1 r
25
25
S 

1  .2 .8
 31.25  31.3 3SF
Geometric sequence
2008 p1 TZ0 #1
Series and sequences
2008 p1 TZ0 #1
(a) Find the 10th term
(b) The infinite sum is
u1, u2,
u3,
u4, …
u
3
2
3
S 

3, 3(.9), 3(.9) , 3(.9) , … u1 = 3
1  r 1  .9
u2 = 3(.9)
3
r=.9
  30
r =.9
u2 3(.9)
.1
u1 = 3
r 
u
3
1
u2 = 3(.9)
r  .9
r =.9
1

The 10th term is
u10  u1 r n-1  3(.9) 9
Geometric sequence
2008 p2 TZ2 #1
2008 p2 TZ2 #1
Series and sequences
(a)
(b)
u1,
u2,
u3, u4, …
3000, -1800, 1080, -648,... u1 = 3000
u2 = -1800
r = -0.6
r =-0.6
u1 = 3000 r  u2  - 1800 n = 10
u2 = -1800
u1 3000
th term is
r =-0.6
The
10
r  .6
u10  u1 r n-1  3000(.6) 9
 30.233088  30.2
3SF
Geometric sequence
2008 p2 TZ2 #1
Series and sequences
(c)
u1,
u2,
u3, u4, …
3000, -1800, 1080, -648,...
r = -0.6
u1 = 3000
u2 = -1800
r =-0.6
The infinite sum is
u1
3000
S 

1  r 1  ( . 6 )
3000

 1875
1 .6
Geometric sequence