Question

# Calculate the 10th term of the infinite series $4,6,8........\infty$.A) 18B) 20C) 22D) 26

Hint: To find the 10th term of the given infinite series we need to first identify the series and then use the formula for its nth term to find the 10th term.
An arithmetic progression is given as: $a,a + d,a + 2d,a + 3d,..........a + (n - 1)d.$ with ‘a’ as the first term and ‘d’ as the common difference. The nth term of this series will be given as:
${a_n} = a + \left( {n - 1} \right)d$.

Complete step-by-step solution:
Now the series given to us is. so the difference between the consecutive terms is given as:
$6 - 4 = 2;8 - 6 = 2...........$
That is the difference between the next term and the previous term is the same for all the consecutive terms and the first term is 4.
Therefore, we can conclude that the given series is an Arithmetic Series with the first term ‘a’=4 and the common difference ‘d’=2.
Now for an Arithmetic Progression the formula to find its nth term is given as:
${a_n} = \left[ {a + \left( {n - 1} \right)d} \right]$
Where ${a_n}$is the nth term, a is the first term n is the number of terms and d is the common difference.
Now for the given series, to find the 10th term we have to take n = 10 and we already have a = 4, d= 2
So putting the values of a, d and n in the formula for nth term we will get ${a_{_{10}}}$as:
${a_{10}} = \left[ {4 + \left( {10 - 1} \right)2} \right] \\ = \left[ {4 + 9 \times 2} \right] \\ = \left[ {4 + 18} \right] \\ = 22 \\$
That is, the 10th term of the given infinite series will be 22.
The given infinite series is an Arithmetic Series and since we had to find the 10th term so we used the formula of the nth term of an Arithmetic Series to get the 10th term and the 10th term is 22.

Hence, the correct answer is option C.

Note: Identifying the series correctly is essential, since if the series is not identified correctly then the whole calculation that follows will be incorrect. The sum of n terms of an Arithmetic Progression is given as: ${S_n} = \dfrac{n}{2}\left( {a + l} \right)$ where a is the first term and l is the last term, the alternative formula for sum of n terms for the same series is given as: ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ . We can use either depending on the question.