Examples. 1 1. Find from (4) the value of n in terms of a, d, and Sn. Ans. 2a [d-2a±√8dSn+ (2a — d)2]. 2. How many terms of 12, 16, 20, .... must be taken to make the sum 208? Ans. 8. 3. How many terms of 3, 2, 6, .... must be taken to make the sum 4481? Ans. 23. 4. Find the expressions for Tn and Sn of the series of Examples 2 and 3. Ans. Tn=8+4n, Sn=2n(n+5) and Sn. Tn. 5. Given T2=3n-1, find the series and the expression for 6. Given Sn=3n2+2n, find the series and the expression for = Ans. Tn 6n-1. 7. Insert ten arithmetic means between 243 and 23. Ans. 223, 203, .... 43. 8. The sum of three numbers in arithmetic progression is 18 and their product is 192; what are the numbers? Ans. 4, 6, 8. 9. Show that the sum of the first n integers is (n+1). T, n 2 ..... 10. In an arithmetic progression T-3, T16=-13. Find Tn, Sn, and the series. Ans. Series, 7, 17, 18, 11. In an arithmetic progression T-2, S.-11. Find Tn, Sn, and the series. Ans. Series, -5, -18, -11,..... 40. Geometric Progressions.-A succession of numbers, each of which is formed by multiplying the one next before it by a given number, is called a geometric progression. The given multiplier is called the common ratio of the progression. For example: 3, 6, 12, 24, etc., form a geometric progression of which the first term is 3 and the common ratio is 2; form a geometric progression of which the first term is 6 and the common ratio is -. The expressions number of terms, last term, and means are used in connection with geometric progressions as with arithmetic. The geometric mean between two numbers is also called their mean proportional, and is equal to the square root of their product. 41. General Term and Last Term.-In any geometric progression, if the first term is a, and the common ratio r, we have by definition: T1=a, T2=ar, T1=ar2, T1=ar3, Tr=arm-1. ..... In a geometric progression of a limited number, n, of terms, the last term, 7, is written: Multiplying Sn by r and subtracting the product from Sn, we have Sn=a+ar+ar2 +ar3 +. +arn-1 .... In case r is greater than 1, it is more convenient to write Equations (1) and (2) furnish the means, when any three of the quantities a, r, n, 1, Sʼn are given, of finding the other two. Examples. 1. Find the sum of 1, 1, 1, etc., to five terms. Ans. 781. 2. The sum of three numbers in geometric progression is 26, their product 216; what are the numbers? Ans. 2, 6, 18. 3. Find S, for the progression 6, -3, +, etc. 42. The Infinite Geometric Progression.-In a geometric progression, if the common ratio is greater than unity (r>1) or less than negative unity (r<-1), the successive terms are increasingly greater in numerical value; if r=1, they are all equal; if r=-1, they are numerically equal, alternating in sign; and if the common ratio is a positive or negative proper fraction (-1<r<1), the successive terms decrease in numerical value. The last case is of special interest. As an example, consider the progression 1 S, is always less than 2, and differs from 2 by 2-1. This differ ence becomes less and less as more terms of the series are taken into the sum, Sn; that is, as n is taken increasingly larger; and it is evident that 1 2-1 may be made less than any conceivable number by making n large enough; that is, by taking enough terms of the series. Such a relation as this is briefly expressed by saying that The limit of Sn as n increases indefinitely is 2. A more usual expression is: The limit of Sn when n is infinite is 2. The formulation for this is: or more briefly, The limit of Sn when n∞ is 2, Sn]p==2. This limit is called the sum of the geometric progression to infinity, or simply the sum of the progression, and is abbreviated S or S. ∞ And if r is less than unity in numerical value, a sufficiently large value of n will make less than any conceivable number so that Sn gets nearer to the value- as more and a α -r more terms are taken, and may be made to differ from by less than any conceivable value by taking enough terms; that is, by making n large enough: For example, the sum to infinity of the geometric progression The nearer r is to 1 in numerical value, the larger the sum of the progression; if r is equal to 1 or numerically greater, the value of S, does not approach any limit as more terms are taken, but increases indefinitely, so that we say that the sum is infinite. S∞∞, if r>1, r<−1, or r=1. 2. How many terms must be taken into Sn in the case of arn 1 be each of the series in Example 1 before the difference tween Sn and S. is less than 0.00001? [Use logarithms.] Ans. 11 and 1280. 5. Find the sums of the repetends 0.037 and 0.09, treating Ans. and 11. them as geometric progressions. 6. Reduce the repetend 2.31243 to the equivalent fraction. Ans. 25 2139 7. Two places, A and B, are a mile apart; a man walks from A toward B, going of the distance; turns, and walks of the way back toward A; turns, and walks of the way back toward his previous turning point. If he keeps this up indefinitely, how many miles will he walk, and how far will he get from A? Ans. 3 mi., mi. |