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

1 Answer
Mar 7, 2016

S= -210

Explanation:

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