Higher Engineering Mathematics, Sixth Edition

Arithmetic and geometric progressions 57

(a) Let the GP bea,ar,ar^2 ,...,arn

The first terma=£100

The common ratior= 1. 08

Hence the second term is

ar=( 100 )( 1. 08 )=£108,

which is the value after 1year,

the third term is

ar^2 =( 100 )( 1. 08 )^2 =£116. 64 ,

which is the value after 2years, and so on.

Thus the value after 10years

=ar^10 =( 100 )( 1. 08 )^10 =£215.89

(b) When £300 has been reached, 300=arn

i.e. 300= 100 ( 1. 08 )n

and 3 =( 1. 08 )n

Taking logarithms to base 10 of both sides gives:

lg3=lg( 1. 08 )n=nlg( 1. 08 ),

by the laws of logarithms

from which,n=

lg3

lg1. 08

= 14. 3

Hence it will take 15years to reach more than

£300.

Problem 20. A drilling machine is to have 6

speeds ranging from 50rev/min to 750rev/ min. If

the speeds form a geometric progression determine

their values, each correct to the nearest whole

number.

Let the GP ofnterms be given bya,ar,ar^2 ,...,arn−^1.
The first terma=50rev/min
The 6th term is given byar^6 −^1 , which is 750rev/min,

i.e. ar^5 = 750

from which r^5 =

750

a

=

750

50

= 15

Thus the common ratio,r=^5

√

15 = 1. 7188

The first term isa=50rev/min

the second term isar=( 50 )( 1. 7188 )= 85 .94,

the third term isar^2 =( 50 )( 1. 7188 )^2 = 147 .71,

the fourth term isar^3 =( 50 )( 1. 7188 )^3 = 253 .89,

the fifth term isar^4 =( 50 )( 1. 7188 )^4 = 436 .39,

the sixth term isar^5 =( 50 )( 1. 7188 )^5 = 750. 06

Hence, correct to the nearest whole number, the 6 speeds

of the drilling machine are50, 86, 148, 254, 436 and

750rev/min.

Now try the following exercise

Exercise 27 Further problems on geometric

progressions

1. In a geometric progression the 5th term is
   9 times the 3rd term and the sum of the 6th
   and 7thterms is 1944. Determine (a) the com-
   mon ratio, (b) the first term and (c) the sum
   of the 4th to 10th terms inclusive.
   [(a) 3 (b) 2 (c) 59022]


2. Which term of the series 3, 9, 27,...is
   59049? [10th]


3. The value of a lathe originally valued at
   £3000 depreciates 15%per annum. Calculate
   its value after 4years. The machine is sold
   when its value is less than £550. After how
   many years is the lathe sold?
   [£1566, 11years]


4. If the populationof Great Britainis 55million
   and is decreasing at 2.4%per annum, what
   will be the population in 5years time?
   [48.71M]


5. 100g of a radioactive substance disintegrates
   at a rate of 3%per annum. How much of the
   substance is left after 11years? [71.53g]


6. If £250 is invested at compound interest of
   6%per annum determine (a) the value after
   15years, (b) the time, correct to the nearest
   year, it takes to reach £750.
   [(a) £599.14 (b) 19 years]


7. A drilling machine is to have 8 speeds rang-
   ing from 100rev/min to 1000rev/min. If the
   speeds form a geometric progression deter-
   mine their values, each correct to the nearest
   whole number.
   [100, 139, 193, 268, 373, 518,
   720, 1000rev/min]