respectively. This indicates that the coefficient of the next term n (n − 1) (n − 2) and in general that the coefficient of the will be 1.2.3 which is in fact the form that our rule (§ 126) would afford for any particular value of r. This affords the following RULE. The (r+ 1)st term of (a+b)" is The form of the coefficient may be easily remembered since the denominator consists of the product of the integers from 1 to r, while the numerator contains an equal number of factors consisting of descending integers beginning with n. For any particular values of n and r we could easily verify the rule by direct multiplication. For the rigorous proof see p. 178. y 12. Find the 8th term in the development of (37-37)10. Solution: The (r+1)st term of (a + b)" is (§ 128) y n = = 10, r + 1 = 8. n(n - 1)... (n r + 1) an-rbr. 1.2 ·· γ In this case we get 2 x 13. Find the 7th term of 14. Find the 6th term of 15. Find the 8th term of 23x3 37y7 33y3 27x7 120.81 y 16 x4 = = 120. 1215 y4 2 x4 1 13 2a√b 3 vb 15 34y4 24x4 = 7 In this exercise any terms beyond those taken would not affect the first three places in the result. 27. In what term of (a + b)20 does a term involving a14 occur? 28. For what kind of exponent may a and b enter the same term with equal exponents? 29. For what kind of exponent is the number of terms in the binomial development even? 30. Find the first three and the last three terms in the development of CHAPTER XIII ARITHMETICAL PROGRESSION 129. Definitions. A series of numbers such that each number minus the preceding one always gives the same positive or negative number is called an arithmetical series or arithmetical progression (denoted by A.P.). The constant difference between any term and the preceding term of an A.P. is called the common difference. The series 4, 7, 10, 13, 8, 61, 5, 31, 10, ... ... is an A.P. with the common difference 3. The series is an A.P. with the common difference - 3. The series 4, 6, 7, 9, is not an A.P. EXERCISES Determine whether the following series are in A.P. If so, find the common difference. 130. The nth term. The terms of an A.P. in which a is the first term and d the common difference are as follows: a, a + d, a + 2d, a + 3d, .... (1) The multiple of d is seen to be 1 in the second term, 2 in the third term, and in fact always one less than the number of the term. If we call 7 the nth term, we have We may follows: also write the series in which is the nth term as 131. The sum of the series. We may obtain a formula for computing the sum of the first n terms of an A.P. by the following THEOREM. The sum s of the first n terms of the series a, a + d, l-d, l is s = a + (a + d) + (a + 2 d) + ··· + (1 − 2 d) + (l − d) +l. (1) Inverting the order of the terms of the right-hand member, s = l + (l− d) + (l − 2 d) + ··· + (a− 2 d) + (a + d) + a. (2) Adding (1) and (2) term by term, ... 2 s = (l + a) + (l + a) + (l + a) + ··· + (l + a) + (l + a) + (l + a) = n (a + 1). Thus n s = 1/2 (a + 1). 132. Arithmetical means. The terms of an A.P. between a given term and a subsequent term are called arithmetical means between those terms. By the arithmetical mean of two numbers is meant the number which is the second term of an arithmetical series of which they are the first and third terms. Thus the a + b arithmetical mean of two numbers a and b is 2 a + b numbers a, 2 The two formulas since the b are in A.P. with the common difference b 2 contain the elements a, l, s, n, d. Evidently when any three are known the remaining two may be found by solving the two equations (I) and (II). |