Higher Engineering Mathematics

54 NUMBER AND ALGEBRA

5. Insert four terms between 5 and 22.5 to form
   an arithmetic progression.
   [8.5, 12, 15.5, 19]


6. The first, tenth and last terms of an arithmetic
   progression are 9, 40.5, and 425.5 respect-
   ively. Find (a) the number of terms, (b) the
   sum of all the terms and (c) the 70th term.
   [(a) 120 (b) 26070 (c) 250.5]


7. On commencing employment a man is paid
   a salary of £7200 per annum and receives
   annual increments of £350. Determine his
   salary in the 9th year and calculate the total
   he will have received in the first 12 years.
   [£10 000, £109 500]


8. An oil company bores a hole 80 m deep. Esti-
   mate the cost of boring if the cost is £30
   for drilling the first metre with an increase
   in cost of £2 per metre for each succeeding
   metre. [£8720]

6.4 Geometric progressions

When a sequence has a constant ratio between suc-
cessive terms it is called ageometric progression
(often abbreviated to GP). The constant is called the
common ratio,r.
Examples include

(i) 1, 2, 4, 8,...where the common ratio is 2 and

(ii)a,ar,ar^2 ,ar^3 ,...where the common ratio isr.

If the first term of a GP is ‘a’ and the common ratio

isr, then

then’th term is:arn−^1

which can be readily checked from the above

examples.

For example, the 8th term of the GP 1, 2, 4, 8,...is

(1)(2)^7 = 128 , sincea=1 andr=2.

LetaGPbea,ar,ar^2 ,ar^3 ,...,arn−^1

then the sum ofnterms,

Sn=a+ar+ar^2 +ar^3 +···+arn−^1 ··· (1)

Multiplying throughout byrgives:

rSn=ar+ar^2 +ar^3 +ar^4

+···+arn−^1 +arn+··· (2)

Subtracting equation (2) from equation (1) gives:

Sn−rSn=a−arn

i.e. Sn(1−r)=a(1−rn)

Thus the sum ofnterms, Sn=

a(1−rn)

(1−r)

which

is valid whenr<1.

Subtracting equation (1) from equation (2) gives

Sn=

a(rn−1)

(r−1)

which is valid whenr> 1.

For example, the sum of the first 8 terms of the GP

1, 2, 4, 8, 16,...is given byS 8 =

1(2^8 −1)

(2−1)

, since

a=1 andr= 2

i.e. S 8 =

1(256−1)

1

= 255

When the common ratiorof a GP is less than unity,

the sum ofnterms,Sn=

a(1−rn)

(1−r)

, which may be

written asSn=

a

(1−r)

−

arn

(1−r)

.

Sincer<1,rnbecomes less asnincreases, i.e.

rn→0asn→∞.

Hence

arn

(1−r)

→0asn→∞. ThusSn→

a

(1−r)

asn→∞.

The quantity

a

(1−r)

is called thesum to infinity,

S∞, and is the limiting value of the sum of an infinite

number of terms,

i.e. S∞=

a

(1−r)

which is valid when− 1 <r<1.

For example, the sum to infinity of the GP

1 +^12 +^14 +···is

S∞=

1

1 −^12

, sincea=1 andr=^12 , i.e.S∞=2.