The middle term of the series

4 + 6 + 8 + ..................... + 196 is

The series is: $4 + 6 + 8 + \ldots + 196$

Here, $196$ is the last term of the series (not the sum).

This is an Arithmetic Progression (AP) where:

First term $(a) = 4$

Common difference $(d) = 6 - 4 = 2$

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Using the $n$th term formula of an AP:

$a_n = a + (n - 1)d$

Where $a_n = 196$, $a = 4$, $d = 2$:

$196 = 4 + (n - 1) \times 2$

$192 = (n - 1) \times 2$

$96 = n - 1$

$n = 97$

There are $97$ terms in total.

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Since $97$ is odd, there is exactly one middle term. For any odd number of terms, the middle term is at the position that has an equal number of terms on both sides of it:

$\text{Middle position} = \dfrac{n + 1}{2}$

$= \dfrac{97 + 1}{2}$

$= \dfrac{98}{2}$

$= 49$

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Using the $n$th term formula:

$a_{49} = 4 + (49 - 1) \times 2$

$= 4 + 96$

$= 100$

Therefore, the middle term of the series is $100$.