Find the general term and then the sum of first $n$ terms of the series: $1 \cdot n+2 \cdot(n-1)+3 \cdot(n-2)+\ldots$
Answer (1)
The general term of the series is found to be T k = k ( n − k + 1 ) .
Expanding the general term gives T k = nk − k 2 + k .
The sum of the first n terms is S n = ∑ k = 1 n ( nk − k 2 + k ) .
Simplifying the summation results in 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 we want to find the general term and the sum of the first n terms.
Finding the General Term Let's find the general term T k of the series. The k t h term is given by T k = k ( n − k + 1 ) for k = 1 , 2 , 3 , … , n .
Expanding the General Term Expanding the general term, we have T k = k ( n − k + 1 ) = nk − k 2 + k .
Finding the Sum of the First n Terms Now, let's find the sum of the first n terms, S n = ∑ k = 1 n T k = ∑ k = 1 n ( nk − k 2 + k ) .
Separating the Summation We can separate the summation as follows: S n = ∑ k = 1 n nk − ∑ k = 1 n k 2 + ∑ k = 1 n k = n ∑ k = 1 n k − ∑ k = 1 n k 2 + ∑ k = 1 n k .
Using Known Formulas Using the formulas for the sum of the first n natural numbers and the sum of the squares of the first n natural numbers, we have ∑ k = 1 n k = 2 n ( n + 1 ) and ∑ k = 1 n k 2 = 6 n ( n + 1 ) ( 2 n + 1 ) .
Substituting the Formulas Substituting these formulas into the expression for S n , we get: S n = n 2 n ( n + 1 ) − 6 n ( n + 1 ) ( 2 n + 1 ) + 2 n ( n + 1 ) .
Simplifying the Expression Simplifying the expression for S n , we have: S n = 2 n 2 ( n + 1 ) − 6 n ( n + 1 ) ( 2 n + 1 ) + 2 n ( n + 1 ) = 6 3 n 2 ( n + 1 ) − n ( n + 1 ) ( 2 n + 1 ) + 3 n ( n + 1 ) .
Further Simplification Further simplification gives: S n = 6 n ( n + 1 ) [ 3 n − ( 2 n + 1 ) + 3 ] = 6 n ( n + 1 ) [ 3 n − 2 n − 1 + 3 ] = 6 n ( n + 1 ) [ n + 2 ] = 6 n ( n + 1 ) ( n + 2 ) .
Final Answer Therefore, the general term is T k = k ( n − k + 1 ) and the sum of the first n terms is S n = 6 n ( n + 1 ) ( n + 2 ) .
Examples
Consider a scenario where you are arranging books on shelves. If each shelf can hold a different number of books, and the number of books decreases linearly from one shelf to the next, this formula helps you calculate the total number of books you can arrange on all the shelves. For instance, if you have 'n' shelves and the number of books on each shelf follows the pattern described by the series, you can quickly determine the total number of books that can be accommodated. This type of calculation is useful in inventory management, resource allocation, and even in designing seating arrangements in a theater or classroom.
