1.4 Sum of the First n Terms of an Arithmetic Progression

1.4 Sum of the First n Terms of an Arithmetic Progression 

(F) Sum of the First n terms of an Arithmetic Progressions
   S n = n 2 [ 2a+( n1 )d ]      S n = n 2 ( a+l )
a = first term
d = common difference
n = the number of term
Sn = the sum of first n terms

Example:
Calculate the sum of each of the following arithmetic progressions.
(a) -11, -8, -5, ... up to the first 15 terms.
(b) 8,   10½,   13,...   up to the first 13 terms.
(c) 5, 7, 9,....., 75 [Smart TIPS: The last term is given, you can find the number of term, n]

Solution:
(a)
11,8,5,.....Find  S 15 a=11, d=8( 11 )=3 S 15 = 15 2 [ 2a+14d ] S 15 = 15 2 [ 2( 11 )+14( 3 ) ]=150

(b)
8,10 1 2 ,13,.....Find  S 13 a=8 d=10 1 2 8= 5 2 S 13 = 13 2 [ 2a+12d ] S 13 = 13 2 [ 2( 8 )+12( 5 2 ) ]=299

(c)
5,7,9,.....,75( The last term l=75 ) a=5 d=75=2 S n = n 2 ( a+l ) S 36 = 36 2 ( 5+75 )=1440 The last term l=75 T n =75 a+( n1 )d=75 5+( n1 )( 2 )=75 ( n1 )( 2 )=70 n1=35 n=36