Question

How do you find the sum of the arithmetic sequence: $2,4,6,8...,n = 20?$

Answer 1

As we know that an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Here in this question we have to find the sum of the arithmetic sequence , we know the formula of the sum of arithmetic sequence is $S_n = {2a+(n-1)d}$ where $n$ is the number of terms, $a =$ first term and $d$ is the common difference. We will first find the common difference between the two terms of the given series and then we will substitute the values in the formula.

Complete step-by-step solution:
As the per given question we have the sequence: $2,4,6,8...$ Here we are provided with the $nth$ term, also we know the formula of $nth$ term is $a + (n - 1)d$.
We have $a = 2,d = 4 - 2 = 2$ and $n = 20$. So by substituting the values in the sum of arithmetic sequence formula: $S_n = {2a+(n-1)d}$
$\Rightarrow {S_{20}} = \dfrac{{20}}{2}\\{ 2 \times 2 + (20 - 1) \times 2\\}$
$\Rightarrow {S_{_{20}}} = 10\\{ 4 + (19 \times 2)\\} = 10 \times 42$. Therefore the value of ${S_{20}} = 420$.

Hence the sum of the given arithmetic sequence is $420$.

Note: We should be aware of the arithmetic sequence and their formula before solving this kind of question. We should carefully substitute the values and solve them. Also we can find the sum of the given arithmetic sequence by adding the first and last term and then divide the sum by two, this is also a formula when the first and the last term is given in the question. And then the sum of the sequence will be the number of terms multiplied by the average number of terms in the sequence. The formula can be written as $S = \dfrac{n}{2}(a + l)$ where the number of terms and $a,l$ are first and last terms.