Elements of algebra : designed for the use of Canadian grammar and common schools . e 3 means and 2 extremes there are inall 5 terms, then r^ - ^ = ^l^^, r* = 81, whence r is evidently = 3,and the series is 4, 12, 36, 108, 324. Ex. 5. Find six numbers in geometrical progression such thatthe sum of the extremes is 99, and the sum of the other fourterms, 90. operation. The sum of the six terms is evidently 99 + 90 = 189. Let a = the first term and y = the common ratio. Then x, xy, xi/^ xy^, xy^, xy^, represent the termsIr^a xy°-x x(y^-l) 189(y-l> 99 . = ye _i . Buta-j/5 ^x =x(y^ + 1) = 99
Elements of algebra : designed for the use of Canadian grammar and common schools . e 3 means and 2 extremes there are inall 5 terms, then r^ - ^ = ^l^^, r* = 81, whence r is evidently = 3,and the series is 4, 12, 36, 108, 324. Ex. 5. Find six numbers in geometrical progression such thatthe sum of the extremes is 99, and the sum of the other fourterms, 90. operation. The sum of the six terms is evidently 99 + 90 = 189. Let a = the first term and y = the common ratio. Then x, xy, xi/^ xy^, xy^, xy^, represent the termsIr^a xy°-x x(y^-l) 189(y-l> 99 . = ye _i . Buta-j/5 ^x =x(y^ + 1) = 99 .\x = ^T^ 2 189(i/-l) 99 21(y2_l) y^-l J/^ + 1 J y^-l y-y^+y^-y + l 21 11 • -y^ + y^ + i - y-^-y^ + y-y+l .-. 2lJ/4-21j/3 + 21?/2- 21y + 21 = 11?/*+ 11^^+ 11 10t/4+ 10^2^ 10 := 21l/^+ 21y 10(y4 + 7/2+l) = 2l7/(!/+l)10(y* + 2!/2 + 1 _ 1/-2) = 21y(y2 + 1)10(y2+i)2_ I0y2 =10(^2^ 1)2- 21y(2/2 21y /9i,A2 (r+i)-Y(;-(y+i)+ 21y ^ ^ 2ly± 29y 50y 5y^ ^ 20 ~ 20 2 = ;16j/2-40J/ + 25 = -16 + 25 = 94j--. = i3;43/ = 5 + 3 = y-299 Therefore the series is 3, 6, 12, 24, 48, 96,. 208 GEOMETRICAL PPcOGRESSlON. [Sect. XI. Ex, G. The sum of four numbers in geometrical progression isequal to the common ratio + 1, and the first term is ,^7. Requiredthe numbers OPERATION. Let r = the common ratio. 1 r r^ ,.3 Then the numbers are jfy, YTj T7> ^^^ It- l+r^-r^ + r^ l+r + r2(l+r) (l+r)(l+r2)Then l+r= ^- = - i4>or J---- - -A- !& _ £4 Exercise LX. Find the last term and the sum of: 1. 3 + 9 + 27+ &c. \o 6 terms. 2. 1 + 2 + 4 + &c., to 9 terms. 3. f+f +f- + &c. to 7 terms. 4. 3-6+12-&c. to 12 terms. 4. 4 - 5 + 6^- &c., to 6 terms. 6. 30 - 15 + 7^ - &c. to 8 the limit to the sum of the infinite series : • + |-^H&c. 8,i + ,l+-,^- + &c. ^ + l|-&c. 10. 64-32 + 16-&c. 11. -623. 12. -7 13. -976 14. -86232. Sum the following series ; 15. 1 + 3 + 9 + &c. to n terms. 16. 2 - I + 2^5 - &c. to n terms. 17.
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