Evolution by Subtraction. 197 



number, and again by the next odd number above that ; add 

 1 to the product, and multiply by the product 

 n(n + l){n(n + 1) + 1} 



already found, and again by 11. 



Let these examples suffice ; we have reached a point at 

 which the formulae begin to have a complicated appearance, 

 though the practical working of them is not beyond the capa- 

 city of a child, and already we have made a great advance on 

 the results of the older methods. We may already perceive 

 from our investigation that the formulae for u n+] —u n are of 

 most use in simplifying results for the lower indices; but that 

 as we go higher we can more safely rely on the formulae for u n 

 which more closely follow the Binomial Theorem, and so, 

 yielding readily to division by such factors as (n + 1), 

 (?i 2 + w + l), &c, often give us combinations of consecutive or 

 nearly consecutive integers, which are far more easily worked 

 in practice than described in rules. 



Now let us tabulate the results of the rules already found. 

 I will give ten terms of each series, so that they may be easily 

 verified by finding their sum in each case to be 10 Wi — 10; that 

 is, a row of nines with a zero at the end. 

 m=l. All the terms are zero. 



m = 2. + 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 +&c. 

 m=3. + 6 + 18 + 36 + 60 + 90 + 126 + 168 + 216 + 270 + &c. 

 m = 4. + 14 + 64 + 174 + 368 + 670 + 1104 + 1694 + 2464 



+ 3438 + &C. 

 m =5. + 30 + 210 + 780 + 2100 + 4650 + 9030 + 15960 



+ 26280 + 40950+ &c. 

 m = 6. + 62 + 664 + 3366 + 11528 + 31030 + 70992 + 144494 



+ 269296 + 468558 + &C. 

 m=7. + 126 + 2058 + 14196 + 61740 + 201810 + 543606 



+ 1273608 + 2685816 + 5217030 + &c. 

 m= 8. + 254 + 6304 + 58974 + 325088 + 1288990 + 4085184 



+ 11012414 + 26269504 + 56953278 + &c. 

 m = 9. + 510 + 19170 + 242460 + 1690980 + 8124570 



+ 30275910 + 93864120 + 253202760 + 612579510 + &c. 

 m = 10. + 1022 + 58024 + 989526 + 8717048 + 50700550 



+ 222009072 + 791266574 + 2413042576 



+ 6513215598 +&c. 

 m = ll. + 2046 + 175098 + 4017156 + 44633820 + 313968930 



+ 1614529686 + 6612607848 + 22791125016 



+ 68618940390 +&c. 



