Find the middle term of the A.P. 6, 13, 20, …… 216.

To find the middle term of an arithmetic progression (A.P.), we need to determine the total number of terms in the sequence.

The formula to find the nth term of an A.P. is given by:

an = a + (n - 1)d

Where:
an is the nth term
a is the first term
n is the position of the term
d is the common difference between consecutive terms

In this case, we have the first term a = 6 and the common difference d = 13 - 6 = 7.

To find the total number of terms, we need to find the value of n in the equation:

216 = 6 + (n - 1) * 7

Simplifying the equation:

216 = 6 + 7n - 7

Combine like terms:

216 = -1 + 7n

Adding 1 to both sides:

217 = 7n

Dividing both sides by 7:

n = 217 / 7
n ≈ 31

So, there are approximately 31 terms in this arithmetic progression.

To find the middle term, we can use the formula:

Middle term = a + ((n + 1) / 2 - 1) * d

Plugging in the values:

Middle term = 6 + ((31 + 1) / 2 - 1) * 7
= 6 + (32 / 2 - 1) * 7
= 6 + (16 - 1) * 7
= 6 + 15 * 7
= 6 + 105
= 111

Therefore, the middle term of the given arithmetic progression is 111.