Calculate S_35 for the arithmetic sequence in which a_5=19 and the common difference is d=-1.3
We have the formula for the n-th term of an arithmetic sequence:
a_n = a_1 + (n-1)d
We are given a_5 = 19 and d = -1.3. Plugging these values into the formula, we have:
19 = a_1 + (5-1)(-1.3)
19 = a_1 + 4(-1.3)
19 = a_1 - 5.2
a_1 = 19 + 5.2
a_1 = 24.2
Now, let's calculate S_35, the sum of the first 35 terms of this arithmetic sequence.
S_n = n/2 * (a_1 + a_n)
S_35 = 35/2 * (24.2 + a_35)
To find a_35, we can use the formula:
a_n = a_1 + (n-1)d
a_35 = 24.2 + (35-1)(-1.3)
a_35 = 24.2 + 34(-1.3)
a_35 = 24.2 - 44.2
a_35 = -20
Now we can substitute the values into the formula for S_35:
S_35 = 35/2 * (24.2 - 20)
S_35 = 35/2 * 4.2
S_35 = 35 * 2.1
S_35 = 73.5
Therefore, S_35 = 73.5.