2. Required the square root of a'-** in an infinite series. 3. Convert Viti into an infinite series. Ans. 1+-+ira &c. 4. Let V x** be converted into an infinite series. % &c. 8 16 128 2 Ans. X C 202 poand divisor, which being multiplied by -, and the product ** 2a the two the root, which must be added to the double of at 2a first terms of the root, for the next compound divisor. And by proceeding thus, the series may be continued as far as is desired. Note. In order to have a true series, the greatest term of the proposed surd must be always placed first. SIMPLE EQUATIONS. AN EQUATION is when two equal quantities, differently 'expressed, are compared together by means of the sign =placed between them. Thus, 52–5=7 is an equation, expressing the equality of the quantities 12-5 and 7 : A simple equation is that, which contains only one un known quantity, in its simple form, or not raised to any power, Thus, atb=c is a simple equation, containing only the unknown quantity x. Reduction of equations is the method of finding the value of the unknown quantity. It consists in ordering the equation so, that the unknown quantity may stand alone on one side of the equation without a coefficient, and all the rest, or the known quantities, on the other side. RULE 1.* Any quantity may be transposed from one side of the equation to the other, by changing its sign. Thus, if x+3=7, then will *==3=4. And, in like manner, if 4*—853x+20, then will 4x 3x=20+8, or x=28. RULE * These are founded on the general principle of performing equal operations on equal quantities, when it is evident, that the results must still be equal ; whether by equal additions, or subs tractions, or multiplications, or divisions, or roots, or powers. RULE 2. If the unknown term be multiplied by any quantity, that quantity may be taken away by dividing all the other terms of the equation by it. Thus, if ax=-ab-a, then will b1. And if 2x+4=16, then will x+2=8, and x=82 6. In like manner, if axtaba=3*, then will *+2= 30 30* and x's 2b. RULE 34 If the unknown term be divided by any quantity, that quantity may be taken away by multiplying all the other terms of the equation by it. X a 2x In like manner, if -2=6+4, then will 2x 3 18+12, and 2x318+12+6=36, or x==18. RULE 4 The unknown quantity in any equation may be made free from surds by transposing the rest of the terms according to the rule, and then involving each side to such a power, as is denoted by the index of the said surd. Thus, if V x2=6, then will Vx=6+2=8, and x=8*=64. And, if w 4x+16=12, then will 4*+16=144, and I 28 4x=144 16-128, or X=32. 4 In In like manner, if V 2x+3+4=8, then will ♡ 2x+3 S4=4 And' 24-4-3=4=54, and 2x264-3=61, or *= =303 RULE 5 If that side of the equation, which contains the unknown quantity, be a complete power, it may be reduced by extracting the root of the said power from both sides of the equation. Thus, if x' +6x+9=25, then will +3=V2555, or X=53= 2. And, if 3** -9=21+3, then will 3* =21+3+9+ 33, and x* = * = 11, or x=V 11. In like manner, if +10=20, then will 2x* +30 = 3 <60, and x' +15=30, or x* =30-15=15, or x=V 15 2x * RULE 6. Any analogy, or proportion, may be converted into an equation, by making the product of the two mean terms equal to that of the two extremes. Thus, if 34 : 16 :: 5 : 10, then will 3* X 10=16X5, and 30%= 80, or x :=25 2x 20* And, if : 0 :: 6 : C; then will ab, and 20% 3 3 zab zab, or x= 20 RULE 7. If any quantity be found on both sides of the equation with the same sign, it may be taken away froin them both; and if every term in an equation be multiplied or divided by the same quantity, it may be struck out of them all. 6 Thus, if 4xtazótaa, then will 4*=b, and x=-. And, if zax+ 5ab=800, then will 3*+56=86, and 8c5b 4 1. Given 54-15=2x+6; to find the value of Å. First, 5x2x=6715 ==7. First, 14x—6x=120-40+ 16 And, therefore, x==120 First, 5ax-2dx=ct36 c+36 5a2d" |