ARITHMETICAL rROq|lESSION. 37I 



2. If the extremes be 3 and 19, and the number of 

 1 erms 9, it is required to find the common difference, und 

 the sum of the whole series. 



Ans. The diiFerence is 2, and the sum Is 99. 



3. A man is to travel from London to a certain place 

 in 12 days, and to go but 3 miles the fir^t day, increasing 

 every day by an equal excess, so that the last day's journey 

 may be 58 miles ; required the daily increase, and the dis- 

 tance of the place from London. 



Ans. Daily Increase 5, distance 366 miles. 



PROBLEM III. 



Given the first tenn^ the last term^ and the common difference^ 

 t^ jind the number cf terms. 



RULE.* 



# - ■ 

 Divide the difference of the extremes by the common 

 difference, and the quotient, increased by i, is the num- 

 ber of terms required. 



EXAMPLES. 



* By tlie last problem, the diifercnce of the extremes, divided 

 by the number of terms less i, gives the common difference ; 

 consequently the same, divided by the common difference, must 

 'uve the number of terms less I ; hence this quotient, augmented 

 by I, must bo the answer to the question. 



In any arithmetical progression, the sum of any two of its 

 terms Is equal to the sum of any other two terms, taken at an equal 

 distance on con^iraiy sides of the former ; or the double of any 

 one term is equal to the sum of any two terms, taken at an equal 

 distance from it on each side. 



The sum of any roimber of terms («) of tlie arithmetical series 

 of odd numbers i, 3, c, 7, 9, Sec. is equal to the square {«^) of 

 that number. 



That 



