By J. Parry Lewis (auth.)

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**Sample text**

Satisfied by n values of the unknown), to be an equation of the form axn +bxn-l +cxn-2 + ... +px +q =0 where a is a non-zero constant, and b, c, ... , p, q are constants which may be positive, negative or zero. - =0 3x5 +7x3 +4x -! =0 8x6 -17x3 =0 -17x 5 =0 Degree4 Degree4 Degree6 Degree5 The important points to notice are: (i) that it is the degree of the highest power of x when the equation is written out in the form indicated in the definition that determines the degree of the equation; and (ii) that if some solutions are difficult to find, or even seem not to exist, it in no way alters the degree of the equation.

Where a is the first term and r is the common ratio. This common ratio is obtained by dividing any term by the preceding one. P. is given by S a(rn -1) r-1 a(1-rn) 1-r SUMMATION AND CONVERGENCE OF SERIES 43 The former of these is the more convenient when r>1, but the latter whenr

P. in the form a, a + d, a + 2d, a +3d, ... where a is the first term and d is the common difference, both a and d being allowed to be positive or negative. Clearly the common difference is found by subtracting one term from the following term. In the above examples it is 50, 3, -4 and -1! respectively. When we examine the series in the form a, a +d, a +2d, a +3d, ... P. is given by S=n(a+l) 2 =~{2a+(n-1)d} Proofs of these results are given in Appendix 8. It follows from the first of these that for the positive integers up to n S=1+2+3+4+ ...