Hence by the formula the sum of the progression is This signifies that we are to find 6 terms in geometrical pro gression such that the first term is 2 and the last 64 243 Substituting in (1) the values a = 2,n=6, 1= The binomial theorem is a formula by means of which any power of a binomial may be expanded into a series. In the following only those cases in which the exponent is a positive integer are considered. Expanding the following by actual multiplication, we have, (a + x)2 = a2 + 2ax + x2 (a + x)3 = a3 + 3a2x + 3ax2 + x3 (a + x) * = a* (a + x)3 = a3 and so on. + 4a3x + 6a2x2 + 4ax3 + 2" + 5a*x + 10a3x2 + 10 a2x3 + 5ax1 + x3 By a consideration of the above we obtain the following laws : I. The number of terms is greater by 1 than the exponent of the binomial. II. The exponent of a in the first term is the same as the exponent of the binomial, and decreases by 1 in each succeeding term. III. The exponent of x in the second term is 1 and increases by 1 in each succeeding term. IV. The coefficient of a in the first term is 1, the coefficient of the second term is the power to which the binomial is raised. In the third it is the power to which the binomial is raised times that quantity minus 1 divided by 1 times 2, and so on as will become apparent from the following: Let n represent the power to which the binomial a +x is to be raised. In the last teim all the factors in both numerator and denominator cancel giving a" (the exponent of a being n―n or 0 and ao = 1 ; hence a disappears from the last term of the series.) Thus for our last term we have " just as we had an in the first term. To illustrate: In (a + x)' the first term was a', the last x1. In (a + x) the first term was a', the last x3. In (a + x)° expanded the coefficient of the second term is 5, that of the third all the factors cancelling. It is to be further noted that the sum of the exponents of the first and second factors in any term of an expansion of a binomial (as a and x) is equal to the power to which the binomial is to be raised. If x is negative the terms of the expansion are alternately positive and negative. Thus, A trinomial may be raised to any power by the binomial theorem if two of its terms be enclosed in a parenthesis and treated as a single term. TO FIND ANY TERM IN AN EXPANSION. The following laws will be found to hold good for any term. in the expansion of (a.+ x)". I. The exponent of x is the number of the term decreased by 1. II. The exponent of a is n minus the exponent of x. III. The last factor of the numerator of the coefficient is greater by 1 than the exponent of a. IV. The last factor of the denominator is the same as the exponent of x. Hence in the 12th term, The exponent of x will be 12 1 or 11. The exponent of a will be n- (121) or n — 11. The last factor of the denominator will be 11. 10. If r represents the required term then in the rth term, 1) or n r + 1. r + 1. The exponent of a will be n (r Let us find the 8th term of (322 - y.— ')" 11. 10.9.8.7.6.5 7 Ans. = 330 (81æ3) (— y −1) — — 26730x3y — " Find the NOTE. The odd terms of an expansion are positive, the even terms negative. |