Question

How do you find the partial sum of \[\sum {1000 - n} \] from n = 1 to 250?

Answer 1

Hint: A Partial Sum is the sum of part of the sequence. The sum of infinite terms is an Infinite Series. Here in this question we have to find the summation of the given term. The sigma notation implies the summation and we have to find summation till the value of n is 20.

Complete step by step solution:
Now consider the given data
\[\sum {1000 - n} \]
This can be written as
\[ \Rightarrow \sum\limits_{n = 1}^{250} {1000 - n} \]
Applying the sigma notation to each term we have
\[ \Rightarrow \sum\limits_{n = 1}^{250} {1000} - \sum\limits_{n = 1}^{250} n \]
By substituting the values for the n from 1 to 250 it will take a long procedure.
As we know that the formula for the summation. The formula \[\sum\limits_{i = 1}^n 1 = n\] and \[\sum\limits_{i = 1}^n i = \dfrac{{n(n + 1)}}{2}\]. By substituting these formulas for the above inequality we have
\[ \Rightarrow 1000n - \dfrac{{n(n + 1)}}{2}\]
Substituting the value of n as 250 to the above inequality we have
\[ \Rightarrow 1000(250) - \dfrac{{(250)(250 + 1)}}{2}\]
On simplifying we get
\[ \Rightarrow 250000 - \dfrac{{(250)(251)}}{2}\]
Simplifying the second term of the above inequality we get
\[ \Rightarrow 250000 - (125)(251)\]
Multiplying the second term we get
\[ \Rightarrow 250000 - 31375\]
On subtracting we get
\[ \Rightarrow 218625\]
Hence we have determined the solution for the given question.

Therefore the answer for the given series is 218625

Note: Partial Sums are sometimes called "Finite Series". To simplify the summation terms we have a standard formula. We must know the formulas which makes the summation problems much easier. The sigma notation implies the summation where summation is the total sum of numbers.