How do you find the sum of the first 14 terms of the arithmetic series 11+7+3+...?

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Answer 1 · 2016-03-07T22:35:28

#S= -210#

There are several formula we need to keep in mind when working with arithmetic series

1) common difference #d = a_(n+1) -a_n#

2) last term : #a_n= a_1+(n-1)d#

3) Arithmetic sum: #S= n/2 (a_1 + a_n)#
An alternative formula is #S=n/2(2a_1+(n-1)d)#

Now we can begin to work this problem

We have an arithmetic series begin at 11, 7, 3 ......

We know that #a_1= color(blue)11#

Let's find the following
#d=7-11 = -4#
#color(green)(n= 14)#
#a_14 = 11+(14-1)(-4)#

#color(red)(a_14= -41)#

Let's substitute it into the sum formula

#S= color(green)(14)/2 ((color(blue)(11))+(color(red)(-41)))#

#S= 7(-30)#

#S= -210#

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If we use the alternative formula, we will still get the same answer
#d= -4 , a_1= 11, n= 14#

#S= 14/2 (2(11)+(14-1)(-4))#

#S= 7(22-52)#

#S= 7(-30)#

#S= -210#