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(F) Sum of the First n Terms of an Arithmetic Progression 

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1.2.3 Sum of the First nTerms of an Arithmetic Progression 

(F) Sum of the First n terms of an Arithmetic Progressions
S n = n 2 [ 2 a + ( n − 1 ) 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 [ 2 a + 14 d ] 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 [ 2 a + 12 d ] S 13 = 13 2 [ 2 ( 8 ) + 12 ( 5 2 ) ] = 299

(c)
5 , 7 , 9 , ..... , 75 ← ( The last term l = 75 ) a = 5 d = 7 − 5 = 2 S n = n 2 ( a + l ) S 36 = 36 2 ( 5 + 75 ) = 1440 The last term l = 75 T n = 75 a + ( n − 1 ) d = 75 5 + ( n − 1 ) ( 2 ) = 75 ( n − 1 ) ( 2 ) = 70 n − 1 = 35 n = 36