Question

In the sequence 2, 4, 6, 8,….. find the \[{{n}^{th}}\] term.

Answer 1

Hint: Find the first term and the common difference of the given series. Thus substitute these values in the equation of A.P. for finding the \[{{n}^{th}}\] term.

Complete step-by-step answer:
Given sequence is the sequence 2, 4, 6, 8…
This is an arithmetic progression (A.P.). A.P. is a sequence of numbers such that the difference between the consecutive terms is constant. Difference here means the second minus the first term. Let us consider d as the common difference of the arithmetic progression.
d = 2^nd term – 1^st term = 4 – 2 = 2
d = 3^rd term – 2^nd term = 6 – 4 = 2
\[\therefore \] Thus we got d = 2.
The first term, a = 2.
We need to find the \[{{n}^{th}}\] term of the given arithmetic progression.
Let \[{{a}_{n}}\] be the \[{{n}^{th}}\] term of the given arithmetic progression.
Thus by using the formula of A.P. to find \[{{n}^{th}}\] term,
\[{{a}_{n}}=a+(n-1)d\]
We know a = 2 and d = 2, substitute and get the value of \[{{a}_{n}}\].
\[{{a}_{n}}=2+(n-1)2=2+2n-2=2n\]
\[\therefore {{a}_{n}}=2n,n\ge 1\]
Thus the last term, i.e. the \[{{n}^{th}}\] term of the given sequence is 2n.
\[\therefore {{a}_{n}}=2n,n\ge 1\].

Note: In the given sequence 2, 4, 6, 8….. there are multiples of 2, i.e. they represent similar patterns. Such a sequence can be expressed in terms of \[{{n}^{th}}\] term.
\[\begin{align}
  & 2\times 1=2 \\
 & 2\times 2=4 \\
 & 2\times 3=6 \\
 & 2\times 4=8 \\
 & 2\times 5=10 \\
\end{align}\]
So we can see that \[{{n}^{th}}\] term can be given as 2n.
Thus \[{{n}^{th}}\] term = 2n.