Question

Find the nth term of the sequence 2,4,6,8,10,…….

Hint: Identify the pattern the subsequent terms are following and then find its general term.

According to the question, the given sequence is 2,4,6,8,10,……
If we observe them carefully, the terms are the nth positive even number from start.
2 is the first positive even number, 4 is the second positive even number, 6 is the third positive even number and so on.
We know that, nth positive even number can be written as $2n$. So, the general term of the given series is:
$\Rightarrow {T_n} = 2n$
Hence, the nth term of the sequence is $2n$.
Note: The sequence is also in arithmetic progression. So, we have
First term, $a = 2$ and common difference, $d = 2$.
We know that the nth term of an A.P. is:
${T_n} = a + \left( {n - 1} \right)d$
Using this formula, the nth term of the given sequence is:
$\Rightarrow {T_n} = 2 + \left( {n - 1} \right) \times 2, \\ \Rightarrow {T_n} = 2 + 2n - 2, \\ \Rightarrow {T_n} = 2n \\$
Here, we have used a different approach but the end result is the same.