Determine whether the finite series is​ arithmetic, geometric,​ both, or neither. If the series is arithmetic or​ geometric, find its sum.

Question

20+19.6+19.2+...+0.4

Answer 1

The series is arithmetic because there is a common difference between each term. The common difference is -0.4.

To find the sum of an arithmetic series, we can use the formula:

Sn = n/2(2a + (n-1)d)

Where:
Sn = sum of the first n terms
n = number of terms
a = first term
d = common difference

In this series, a = 20, d = -0.4, and we need to find n.

To find n, we can use the formula for the nth term of an arithmetic series:

an = a + (n-1)d

In this series, we need to find the value of n when an = 0.4, a = 20, and d = -0.4.

0.4 = 20 + (n-1)(-0.4)
0.4 = 20 - 0.4n + 0.4
0.4n = 20
n = 50

Now that we have the value of n, we can find the sum of the series:

Sn = 50/2(2(20) + (50-1)(-0.4))
Sn = 25(40 + 49(-0.4))
Sn = 400 + 25(-19.6)
Sn = 400 - 490
Sn = -90

Therefore, the sum of the series 20+19.6+19.2+...+0.4 is -90.