intend a sequence of number is arithmetic (that is, it increases or to reduce by a consistent amount each term), and also you desire to find the sum of the first n terms.

denote this partial amount by S n . Then

S n = n ( a 1   +   a n ) 2 , where n is the number of terms, a 1 is the an initial term and a n is the critical term.

The amount of the an initial n terms of an arithmetic succession is referred to as an arithmetic series .


example 1:

find the sum of the an initial 20 regards to the arithmetic series if a 1 = 5 and a 20 = 62 .

S 20 = 20 ( 5   +   62 ) 2 S 20 = 670


instance 2:

find the sum of the an initial 40 terms of the arithmetic succession 2 , 5 , 8 , 11 , ⋯ .

first find the 40 th term:

a 40 = a 1 + ( n − 1 ) d                 = 2 + 39 ( 3 ) = 119

Then uncover the sum:

S n = n ( a 1   +   a n ) 2 S 40 = 40 ( 2   +   119 ) 2 = 2420


example 3:

discover the sum:

∑ k = 1 50 ( 3 k + 2 )

first find a 1 and also a 50 :

a 1 = 3 ( 1 ) + 2 = 5 a 50 = 3 ( 50 ) + 2 = 152

Then uncover the sum:

S k = k ( a 1   +   a k ) 2 S 50 = 50 ( 5   +   152 ) 2 = 3925


view also: sigma notation the a series and nth term of one arithmetic succession


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