The middle term of the series
4 + 6 + 8 + ....... + 196 is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 100
The terms 4, 6, 8, …, 196 form an arithmetic progression with first term 4 and common difference 2. With an odd number of terms \(n\), the middle term is the \(\frac{n+1}{2}\)th.
Number of terms: from 4 to 196 there are \(\frac{196-4}{2}\) steps of 2, and the first term adds one more: \(\displaystyle\frac{196-4}{2}+1=97\)
The middle term is the \(\displaystyle\frac{97+1}{2}=49\)th term
The 49th term lies 48 steps of 2 after the first term: 49th term \(=4+48\times2=100\)
Shortcut:
In an AP with an odd number of terms, the terms are spread evenly about the middle, so the middle term is the average of the first and last terms
Middle term \(\displaystyle=\frac{4+196}{2}=100\)