6 MATHEMATICAL THEORY OF INVESTMENT 4. The first three terms of a progression are 3, 6, and 9. Find the num- ber of terms when the sum of all the terms is 234. 5. Suppose that in Example 4 the sum were 225 and the remaining data the same. What would be the value of TO ? What is the meaning of the result ? 6. The first three terms of a progression are 3, 6,12. Find the sum of 10 terms. 7. An elastic ball is dropped from a height of 20 feet, and each time it strikes the ground it rebounds to one half the distance through which it fell. How far will it have traveled when it strikes the ground for the tenth time? 8. Find the sum of the multiples of 3 that lie between 200 and 400. 9. A body falling from rest falls approximately 16.1 feet the first second, 48.3 feet the second, 80.5 feet the third, and so on. How far will it fall in 10 seconds? 10. The value of the timber in a certain forest increases at the rate of 4% annually. If it was worth $10,000 at the beginning of a five-year period, how much will it be worth at the end of the period ? ,_ .^ . 11. Find the sum of five terms of the series . - >, ——1—-, 5-2V6, 9V3-11V2, . V3+V2 12. Prove that three numbers in arithmetical progression cannot be in geometrical progression unless the common difference is zero. 13. Prove that three numbers in geometrical progression cannot be in arithmetical progression at the same time unless the ratio is + 1. 3. Geometrical progressions in which the number of terms increases indefinitely. In every problem in progressions so far considered the number of terms has been finite. There are, how- ever, many important problems whose solution requires the con- cept of a decreasing geometrical progression in which the number of terms is indefinitely increased. An example will help to make the concept clear. Suppose a stick 2 feet long is cut into two equal parts, and one of these is cut into two equal parts, and so on, one of the two parts into which a given piece is cut being bisected each time. In this way a series of shorter sticks is obtained whose lengths are 1, ^, y y ^g, and so on, each fraction having for its PROGRESSIONS 7 denominator a power of 2. These terms, as far as they go, form a geometrical progression whose ratio is ^ and whose nth term is given by /IV"1 1 ^:=1U =on^i-
The sum of n terms will be 1
{2/2-1
?-1 „ 1 8=-;————=2-^
T_l 2-1 2 Consider now what happens if the process of bisection is car- ried on indefinitely. Two things are clear from the nature of the problem: First, the length of the last piece becomes indefi- nitely small, or, to use a customary phrase, the length of the last piece tends toward zero as a limit. In the second place, however far the process is carried on, the sum of all the pieces cut off can never exceed 2, though it can be made to approach as near to 2 as we please. This is seen from the expression just found for s, viz. . l
2"-1
This relation may be written 2-8=
2"-1
from which form it can be seen that the right member becomes small at will when n is increased, and consequently the difference 2—8 may be made as small as we please. In such a case it is customary to say that the limit of 2 — 8 is equal to 0, where the term limit is used in accordance with a definition to be formulated in the next chapter, and the relation is written lim(2-8)=0. n=oo From this relation we deduce, by Theorem IV of § 4, lim 8=2, n=<» and thus are brought back to the length of the stick in its original condition. 8 MATHEMATICAL THEORY OF INVESTMENT To generalize these notions for any geometrical progression whose ratio is numerically less than unity, consider the sum «„ = a + ar + ar2 + ... + ar"~1, where the subscript n is used to emphasize the fact that the sum of n terms is under consideration. By (Z>) of §. 2, a — ar"
" 1-r ,., , .„ a ar" which may be written «„ = .—— — .——» a ar" or ^———8„=:;—— 1 _ ,- ° 1 _ r If in this equation r be taken numerically less than unity, r" will be smaller than r itself/and as n increases, r", and conse- j'»A<^* quently :,——, will tend toward zero as a limit. According to the notation used in the stick problem above, T / a \ ,. ar" „ i^T^-8^11111!—^0' and finally lim^=,—— fE~) n=oo 1 — T It is important to note carefully that equation (.£') does not state that «„ is equal to ,——. The equation means just what it says, namely, :.—— is the limit toward which «„ is tending, and that by taking n large enough the difference between »„ and —— can be made as small as we please. Its importance lies in the fact that it furnishes a direct means for the solution of all prob- lems like the stick problem. In that problem a = 1 and r = A. Substituting these values in equation (-£'), lim «„ = ——— = 2. - i-| PROGRESSIONS 9 Another example is the determination of the limit toward which a repeating decimal like .333 tends. The repeating decimal may be written in the form .3+.03+.003+---, from which it is readily seen that the terms are in geometrical progression, with the first term a = .3 and the ratio r == .1. If these values be substituted in the right member of equation (-£'), the result is '3 1 l^l^S' which is, as we know, the common fraction that gives rise to the repeating decimal .333 In the same way we may find the com- mon fraction which is the equivalent of any repeating decimal. For example, the repeating decimal .23459459 is equivalent to the decimal .23 increased by the geometrical progression .00459 +.00000459 + , whose ratio is .001; consequently, oo^n... _o, ,00459 217
.23459459 =.23 +
1-.001 925
A geometrical progression in which the ratio is numerically less than unity and in which the number of terms is increased indefinitely is called an infinite geometrical progression. EXAMPLES 1. Find the limit of the sum for each of the following infinite geomet- rical progressions: (a) 2 + .5 + .125 + ..., (b) V2 +1 + 1 + , v / -N/2 (c) V2-H+1+V2-1+--.. 2. "What common fraction is the equivalent of the repeating decimal .237237 ? 3. Find the c