Higher Engineering Mathematics, Sixth Edition

56 Higher Engineering Mathematics

The sum of 9 terms,

S 9 =

a( 1 −rn)

( 1 −r)

=

72. 0 ( 1 − 0. 89 )

( 1 − 0. 8 )

=

72. 0 ( 1 − 0. 1342 )

0. 2

=311.7

Problem 16. Find the sum to infinity of the

series 3, 1,^13 ,...

3, 1,^13 ,...is a GP of common ratio,r=^13

The sum to infinity,

S∞=

a

1 −r

=

3

1 −^13

=

3

2

3

=

9

2

= 4

1

2

Now try the following exercise

Exercise 26 Further problems on geometric

progressions

1. Find the 10th term of the series 5, 10, 20,
   40,... [2560]


2. Determine the sum of the first 7 terms of the
   series^14 ,^34 ,2^14 ,6^34 ,... [273.25]


3. The first term of a geometric progression is 4
   and the 6th term is 128. Determine the 8th and
   11th terms. [512, 4096]


4. Find the sum of the first 7 terms of the
   series 2, 5, 12^12 ,...(correct to 4 significant
   figures). [812.5]


5. Determine the sum to infinity of the series 4,
   2, 1,... [8]


6. Find the sum to infinity of the series 2^12 ,− 114 ,
   5
   8 ,...

[

(^123)
]

6.6 Further worked problems on

geometric progressions

Problem 17. In a geometric progression the sixth

term is 8 times the third term and the sum of the

seventh and eighth terms is 192. Determine (a) the

common ratio, (b) the first term, and (c) the sum of

thefifthtoeleventhterms,inclusive.

(a) Let the GP bea,ar,ar^2 ,ar^3 ,...,arn−^1

The 3rd term=ar^2 and the sixth term=ar^5

The 6th term is 8 times the 3rd.

Hencear^5 = 8 ar^2 from which,r^3 =8,r=^3

√

8

i.e.the common ratior= 2.

(b) The sum of the 7th and 8th terms is 192. Hence

ar^6 +ar^7 =192.

Since r= 2 , then 64a+ 128 a= 192

192 a= 192 ,

from which,a,thefirstterm,= 1.

(c) The sum of the 5th to 11th terms (inclusive) is

given by:

S 11 −S 4 =

a(r^11 − 1 )

(r− 1 )

−

a(r^4 − 1 )

(r− 1 )

=

1 ( 211 − 1 )

( 2 − 1 )

−

1 ( 24 − 1 )

( 2 − 1 )

=( 211 − 1 )−( 24 − 1 )

= 211 − 24 = 2048 − 16 = 2032

Problem 18. A hire tool firm finds that their

net return from hiring tools is decreasing by

10% per annum. If their net gain on a certain tool

this year is £400, find the possible total of all future

profits from this tool (assuming the tool lasts for

ever).

The net gain forms a series:

£400+£400× 0. 9 +£400× 0. 92 +···,

which is a GP witha=400 andr= 0 .9.

The sum to infinity,

S∞=

a

( 1 −r)

=

400

( 1 − 0. 9 )

=£4000=total future profits

Problem 19. If £100 is invested at compound

interest of 8% per annum, determine (a) the value

after 10 years, (b) the time, correct to the nearest

year, it takes to reach more than £300.