Find the middle term of the AP 6, 13, 20, ..., 216.
Middle Term of an AP is found by first determining the total number of terms (31) and then calculating the 16th term, giving the middle term as 111.
Find the middle term of the AP 6, 13, 20, ..., 216.
The given AP is 6, 13, 20, ..., 216.
First term, a = 6
Common difference, d = 13 − 6 = 7
Suppose there are n terms in the given AP. Then,
$a_{n}=216$
$\Rightarrow 6+(n-1) \times 7=216 \quad\left[a_{n}=a+(n-1) d\right]$
$\Rightarrow 7(n-1)=216-6=210$
$\Rightarrow n-1=\frac{210}{7}=30$
$\Rightarrow n=30+1=31$
Thus, the given AP contains 31 terms.
∴ Middle term of the given AP
$=\left(\frac{31+1}{2}\right)$ th term
= 16th term
= 6 + (16 − 1) × 7
= 6 + 105
= 111
Hence, the middle term of the given AP is 111.
Frequently Asked Questions
Find answers to common questions.
What is the middle term of the AP 6, 13, 20, …, 216?
The middle term is 111. The AP has 31 terms (n = 31, which is odd), so the middle term falls at the 16th position. Using aₙ = a + (n−1)d: a₁₆ = 6 + 15 × 7 = 6 + 105 = 111. This is the only middle term since n is odd.
How do you find the number of terms in an AP when the first and last terms are given?
Use the formula aₙ = a + (n − 1)d. Substitute the last term for aₙ, first term for a, and common difference for d. Solve for n. For AP 6, 13, 20, …, 216: 216 = 6 + (n−1)×7, which gives n = 31. Always check that n is a positive integer — if it is not, the given value is not actually a term of the AP.
How do you find the middle term position in an AP?
If the AP has n terms and n is odd, the middle term is at position (n + 1)/2. If n is even, there are two middle terms at positions n/2 and (n/2) + 1. Finding n comes first — use the nth term formula with the last known term. The position formula only applies after you have confirmed the value of n.
What if the AP has an even number of terms — is there still a middle term?
When n is even, the AP has two middle terms, not one. They sit at positions n/2 and (n/2 + 1). Some problems ask for the average of these two values. For example, if n = 30 in an AP, the 15th and 16th terms are both considered middle terms. Read the question carefully to know whether a single value or both are required.
Why is verifying the middle term important in board exams?
Verification shows the examiner your working is consistent and earns method marks even if a small arithmetic error exists elsewhere. The fastest check: confirm that the terms immediately before and after the middle term each differ from it by exactly d (the common difference). For this AP, a₁₅ = 104 and a₁₇ = 118, both differing from 111 by 7. Verified in seconds.