Answer :
Let's try to find a pattern to the sequence. You can see that we start from 1.5 and subtract 0.05 to generate the next element with each step. So, if we start from [tex] a_0=1.5 [/tex], the general formula for the n-th element is [tex]a_n=1.5-0.05n[/tex]
So, the sum of the first 20 terms is
[tex] \displaystyle \sum_{n=0}^{19} (1.5-0.05n) [/tex]
We can split the sum:
[tex] \displaystyle \sum_{n=0}^{19} 1.5-\sum_{n=0}^{19} (0.05n) = \sum_{n=0}^{19} 1.5-0.05\sum_{n=0}^{19} n [/tex]
The first sum is independent of n, so we're just summing 1.5 for 20 times:
[tex] \displaystyle \sum_{n=0}^{19} 1.5= 1.5\cdot 20 = 30[/tex]
The second sum is 0.05 times the sum of the first 19 integers. The sum of the first k integers is given by
[tex] \dfrac{k(k+1)}{2} [/tex]
So, the sum of the first 19 integers is
[tex] \dfrac{19\cdot 20}{2}=190 [/tex]
and 0.05 times this sum is
[tex] 190\cdot 0.05 = 9.5 [/tex]
So, the sum of the first 20 elements of the sequence is given by
[tex]\displaystyle \sum_{n=0}^{19} 1.5-0.05\sum_{n=0}^{19} n = 30 - 9.5 = 20.5 [/tex]