#### To determine

**To calculate:** The sum of the series.

#### Answer

The sum of the series is ∑n=1∞an=16.

#### Explanation

**Given:**

The sequence is an=(5−n)an−1. (1)

The first term of the sequence is a1=1.

**Calculation:**

Obtain the sum of the series.

Substitute 2 for *n* in equation (1),

a2=(5−2)a2−1=3a1

Substitute 1 for a1 in the above equation,

a2=3⋅1=3

Thus, the second term of the sequence is a2=3.

Substitute 3 for *n* in equation (1),

a3=(5−3)a3−1=2a2

Substitute 3 for a2 in the above equation,

a3=2⋅3=6

Thus, the third term of the sequence is a3=6.

Substitute 4 for *n* in equation (1),

a4=(5−4)a4−1=1a3=a3

Thus, the fourth term of the sequence is a4=6.

Substitute 5 for *n* in equation (1),

a5=(5−5)a5−1=0a4=0

Thus, the fifth term of the sequence is a5=0.

Since a5=0 then all succeeding terms a6,a7,a8,... are equal to zero.

That is, a6=a7=a8=...=0.

Therefore, the sum of the series is

∑n=1∞an=∑n=14an+∑n=5∞an=∑n=14an+0=∑n=14an=a1+a2+a3+a4

Substitute the values of a1,a2,a3 and a4 to get

∑n=1∞an=1+3+6+6=16

Therefore, the sum of the series is ∑n=1∞an=16.