Find the general term and then the sum of the first [tex]$n$[/tex] terms of the series: [tex]$1 \cdot n+2 \cdot(n-1)+3 \cdot(n-2)+\ldots$[/tex]
Answer (1)
The general term of the series is found to be T k = ( n + 1 ) k − k 2 .
The sum of the first n integers is ∑ k = 1 n k = 2 n ( n + 1 ) .
The sum of the squares of the first n integers is ∑ k = 1 n k 2 = 6 n ( n + 1 ) ( 2 n + 1 ) .
The sum of the first n terms of the series is S n = 6 n ( n + 1 ) ( n + 2 ) .
S n = 6 n ( n + 1 ) ( n + 2 )
Explanation
Understanding the Problem We are given the series 1"." n + 2"." ( n − 1 ) + 3"." ( n − 2 ) + … and asked to find the general term and the sum of the first n terms.
Finding the General Term Let's analyze the terms of the series. The k -th term can be written as T k = k ( n − k + 1 ) , where k ranges from 1 to n .
Simplifying the General Term Now, let's simplify the general term:
T k = k ( n + 1 ) − k 2 = ( n + 1 ) k − k 2
Expressing the Sum Next, we need to find the sum of the first n terms, which we'll denote as S n . This can be expressed as:
S n = ∑ k = 1 n T k = ∑ k = 1 n [( n + 1 ) k − k 2 ]
Separating the Summation We can separate the summation into two parts:
S n = ( n + 1 ) ∑ k = 1 n k − ∑ k = 1 n k 2
Using Known Formulas Now, we'll use the formulas for the sum of the first n integers and the sum of the first n squares:
∑ k = 1 n k = 2 n ( n + 1 )
∑ k = 1 n k 2 = 6 n ( n + 1 ) ( 2 n + 1 )
Substituting the Formulas Substitute these formulas into the expression for S n :
S n = ( n + 1 ) 2 n ( n + 1 ) − 6 n ( n + 1 ) ( 2 n + 1 )
Simplifying the Expression Now, let's simplify the expression for S n :
S n = 6 n ( n + 1 ) [ 3 ( n + 1 ) − ( 2 n + 1 )] = 6 n ( n + 1 ) [ 3 n + 3 − 2 n − 1 ] = 6 n ( n + 1 ) ( n + 2 )
Final Answer Therefore, the general term is T k = ( n + 1 ) k − k 2 and the sum of the first n terms is S n = 6 n ( n + 1 ) ( n + 2 ) .
Examples
This type of series can be used to model scenarios where the contribution of each element decreases linearly. For example, consider a project where each day, fewer resources are available. If on day 1, n resources are available, on day 2, n − 1 resources are available, and so on, and the daily output is proportional to the day number times the available resources, then the sum of the series represents the total output of the project. Understanding such series helps in resource allocation and project planning.
