Series Definition & Meaning

QUESTION 1
(a)
(i) Series C
(ii) Series B
(iii) Series H
(b)
(i) Series C the first term is 6
(ii) Series C: Common difference (d) = 7 – 6 = 8 – 7 = 9 – 8 = 10 – 9 = 1
(iii) The nth term, an is given by
an=a1 + ( n−1 ) d
Where a1 is the first term.
For series C, the 10th term is
a10=6+9 d =15
(iv) The sum of the first 10 terms is given by
¿ n
2 [ 2 a+ ( n−1 ) d ]
For series C, the sum is
¿ 10
2 [ 2(6)+ ( 10−1 ) ]=105

(c)
(i) Series B the first term is 4
(ii) Series B: Common ratio (r) is
¿ 2
4 =1
2 = 0.5
1 = 0.25
0.5 =0.5
(iii) The nth term, an is given by
an=a1 rn−1
Where a1 is the first term.
For series B, the 8th term is
a8=4 ( 0.5 )7= 1
32 =0.03125
(iv) The sum of the first 8 terms is given by
Sn= a1 (rn−1)
r−1 , for r >1
Or
Sn= a1 (1−rn )
1−r , for r <1
For series B, the sum is
¿ 4 (1− ( 0.5 ) 8)
1−0.5 =7.96875

(v) By geometric series test:
Series B converges since the absolute value of the common ratio is less than unity i.e.
r =1
2 which is ¿ 1
(d)
Series H is not Arithmetic Progression (AP) since there exist no common difference (d)
between the consecutive terms i.e.
Series H: (4 – 1) ≠ (9 – 4) ≠ (16 – 9) ≠ (25 -16)
Series H is not Geometric Progression (AP) since there exist no common ratio (r)
between the consecutive terms i.e.
For series H: 4/1 ≠ 9/4 ≠ 16/9 ≠ 25/14
QUESTION 2
The tunneling machine follows an Arithmetic Progression (AP) in drilling. This progression is
defined by the model:
Cost of drilling=X + ( 500−1 ) Y
For X = 1200 and Y =100, the total cost of drilling is
¿ 1200+100 ( 500−1 )
¿£ 51100

QUESTION 3
a)
From the general relation of Geometric Progression T n=a r n−1
First term: T 1=a=X=14
Fifth term: T 5=a r 4=224
From the fifth term expression r =( 224
a ) 1
4
But a=14 from the first term expression
This implies that r =( 224
14 )1
4 =2
∴Common ratio=2
b)
Term Expression Speed
First T 1=a r0 =14 ( 2 )0 14 rpm
Second T 2=a r1=14 ( 2 )1 28 rpm
Third T 3=a r2 =14 ( 2 )2 56 rpm
Fourth T 4=a r3=14 ( 2 )3 112 rpm
Fifth T 5=a r 4=14 ( 2 ) 4 224 rpm