Question

How do you find the partial sum of \[\sum {\left( {2n + 5} \right)} \] from $n = 1 to 20$?

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
\[\sum {\left( {2n + 5} \right)} \]
This can be written as
\[ \Rightarrow \sum\limits_{n = 1}^{20} {2n + 5} \]
Applying the sigma notation to each term we have
\[ \Rightarrow \sum\limits_{n = 1}^{20} {2n} + \sum\limits_{n = 1}^{20} 5 \]
Where 2 and 5 are constant take outside from the sigma
\[ \Rightarrow 2\sum\limits_{n = 1}^{20} n + 5\sum\limits_{n = 1}^{20} 1 \]
By substituting the values for the n from 1 to 20 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
\[ \Rightarrow 2\dfrac{{n\left( {n + 1} \right)}}{2} + 5n\]
Substituting the value of n as 20 to the above inequality we have
\[ \Rightarrow 2\dfrac{{20\left( {20 + 1} \right)}}{2} + 5\left( {20} \right)\]
On simplifying we get
\[ \Rightarrow 420 + 100\]
\[ \Rightarrow 520\]
Hence, the partial sum of \[\sum {\left( {2n + 5} \right)} \] from n = 1 to 20 is 520 or

The sum of \[\sum\limits_{n = 1}^{20} {\left( {2n + 5} \right)} \] is 520.

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 or the above question is also solve by using the formula \[\sum\limits_n^{} {\dfrac{n}{2}\left( {{a_1} + {a_n}} \right)} \] means it's the number of terms times the average of the first and last term.