How do you find the \[{{25}^{th}}\] partial sum of the arithmetic sequence \[{{a}_{1}}=100,{{a}_{25}}=220\]?

Hint: We are asked to find the \[{{25}^{th}}\] partial sum of the arithmetic sequence and we are given the first term and the \[{{25}^{th}}\] term. We will use the formula of the partial sum of the arithmetic sequence, which is, \[{{S}_{n}}=\dfrac{n}{2}({{a}_{1}}+{{a}_{n}})\], here ‘n’ is the number of terms under consideration. We will substitute the given values of \[{{a}_{1}}=100,{{a}_{25}}=220\] in the above formula, on solving which, gives us the value of the \[{{25}^{th}}\] partial sum of the arithmetic sequence.

Complete step by step solution:
According to the given question, we are given \[{{a}_{1}}=100,{{a}_{25}}=220\] of a arithmetic sequence and we have to find the \[{{25}^{th}}\] partial sum of the given arithmetic sequence.
Arithmetic sequence can be defined as a sequence of numbers which has the difference between any two consecutive numbers.
The partial sum of an arithmetic sequence can be written as,
\[{{S}_{n}}=\dfrac{n}{2}({{a}_{1}}+{{a}_{n}})\]
Where ‘n’ is the number of terms under consideration.
We will now substitute the values given in the question in the above formula for the partial sum of an arithmetic sequence, we have,
\[\Rightarrow {{S}_{25}}=\dfrac{25}{2}({{a}_{1}}+{{a}_{25}})\]
Substituting the values in the above expression, we have,
\[\Rightarrow {{S}_{25}}=\dfrac{25}{2}(100+220)\]
Adding up the terms in the bracket and on solving further, we get the expression as,
\[\Rightarrow {{S}_{25}}=\dfrac{25}{2}(320)\]
Now, we will divide 320 by 2 and the resultant we get is multiplied by 25, we het the new expression as,
\[\Rightarrow {{S}_{25}}=25\times 160\]
\[\Rightarrow {{S}_{25}}=4000\]
Therefore, \[{{S}_{25}}=4000\]

Note: The correct formula of the sum of the arithmetic sequence should be applied. Also, the substitution of the values should be done step wise and correctly as well. The question should be read carefully as to what exactly the question is asking. As there are other formulae for the sum of an arithmetic sequence and so has a possibility of getting the question wrong. So based on what all values are given, accordingly use the most suitable one to find the required.