We are given a finite arithmetic series in summation notation and want to find its sum.
∑^(12)_(k=4) (- 2k+5)
To calculate the sum, we need to find the first and last terms. Let's substitute 1 and 12 for k in - 2k+5.
a_k=- 2k+5
a_1=- 2( 1)+5
a_(12)=- 2( 12)+5
a_1=3
a_(12)=- 19
Now that we know a_1=3 and a_(12)=- 19, we will calculate the sum of the series substituting a_1=3, a_k=- 19, and k=12 in the formula for the sum of a finite arithmetic series.
The sum of the first 12 terms of the series is - 96. However, if we pay close attention to the given summation notation, we can see that our series starts at k= 4. This means that we only want to calculate the sum from k=4 to k=12.
∑^(12)_(k= 4) (- 2k+5)
Since we already found the sum of all the terms, we will now calculate the sum of the first three terms and find the difference. To find the sum of the first three terms, we need to calculate the third term, a_3, and then we can once again use the formula for the sum of an arithmetic series.
