The sum of given arithmetic sequence: sum n=1 80 2n - 5=6080

Formula used:
The formula for the summation of a polynominal with degree 1 is, ${\displaystyle \sum k=1nk=n2(n+1)}$ ...(1)
The formula for the summation of a constant is, ${\displaystyle \sum k=1nc=cn}$ ...(2)
Calculation:
To find the sum of the arithmetic sequence, i.e, to find,
${\displaystyle Sn=\sum n=180(2n-5)}$
Write ${\displaystyle \sum n=1802n-5}$ as follows,
${\displaystyle \sum n=1802n-5=2\sum n=180n+\sum n=180-5}$
Substitute the values of n in equation (1).
We get,
${\displaystyle 2\sum n=180n}$
${\displaystyle =2\left(80(80+1)2\right)}$
${\displaystyle =2(80\times 812)}$
${\displaystyle =6480}$
Now to find the ${\displaystyle \sum n=180-5}$
Substitute the values of n in equation (2).
We get,
${\displaystyle \sum n=180-5}$
${\displaystyle =-5\left(80\right)}$
${\displaystyle =-400}$
Then,
${\displaystyle 2\sum n=180n+\sum n=180-5=6480-400}$
${\displaystyle =6080}$
Hence
${\displaystyle \sum n=1802n-5=6080.}$