110 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Assume that any successiv two of these numbers, not in-
cluding the last, say s;_2 and s;_;, where 2 < k < #n + 1, are
positiv.

Then, since gy, is positiv, ¢iSi—1 is positiv.

Hence sx_2 + ¢iSi—1 1s positiv.

That 1s, s 1s positiv.

Thus, if any two successiv numbers in the series sy, $3, §3, * * *,
Sa, not including the last, are positiv, so is the next number.

Therfor by mathematical induction all the numbers in the
series are positiv.

Hence s;, is positiv for all possible values of k, except & = 0,
for which sy is zero.

Similarly we can show by mathematical induction that all

 
   
 
     
   
   

the numbers in the series fs, f3, t4, - -+, t, are positiv and that
Ir is positiv for all possible values of %, except in the cases
when either k = — 1, or k = 1 and @ < b, in which cases #
1s zero.
Ex. 1
¢ check
39 1 PRl 24 = 39
24 0 1[0X39— 1X24=—24

 

ESH L5 ISRAEE39 =T 124 =" =1
Ol i) =l 211X%X39— 2X24=-— 9
04 -2 312X39— 3X24= 6
sl elreS DRSS9 b = — 3
a2 18 3|1 8X39—-13X24= 0

  

Ex. 2.
check

 

1X24—-0X6= 24
0X24—-1X6=—6

 

1X24—-4X6= 0