Given:
An AP with 31 terms
16th term = m

To find:
Sum of all the terms of the AP

Solution:
We know that the sum of an AP is given by the formula:

Sn = n/2[2a + (n-1)d]

where,
Sn = sum of n terms
a = first term
d = common difference

We need to find the sum of all the 31 terms of the AP.
So, n = 31

We know the 16th term of the AP is m.
So, a + 15d = m ---(1)

Also, we know that the 31st term of the AP is a + 30d.
So, using the formula for nth term of an AP, we get:

a + 30d = a + (n-1)d
=> 30d = (n-1)d
=> n = 31

Therefore, the sum of all the terms of the AP is:

S31 = 31/2[2a + (31-1)d]

We can simplify this expression using equation (1) as:

S31 = 31/2[2(m-15d) + (31-1)d]
=> S31 = 31/2[2m - 28d]
=> S31 = 31(m - 14d)

But we don't know the value of d, so we need to eliminate it.

Subtracting equation (1) from a + 16d = m, we get:

d = (m-a)/16

Substituting this value of d in the expression for S31, we get:

S31 = 31(m - 14((m-a)/16))
=> S31 = 31/16[32m - 28a - 14m + 14a]
=> S31 = 31/16[18m - 14a]
=> S31 = 31/8[9m - 7a]

Now, we need to find the value of a, the first term of the AP.

Using equation (1), we get:

a = m - 15d
=> a = m - 15((m-a)/16)
=> a = (31m - 15a)/16
=> 16a = 31m
=> a = 31m/16

Substituting this value of a in the expression for S31, we get:

S31 = 31/8[9m - 7(31m/16)]
=> S31 = 31/8[9m - (217m/16)]
=> S31 = 31/8[(144m/16) - (217m/16)]
=> S31 = 31/8[-73m/16]
=> S31 = -73m/2

Therefore, the sum of all the terms of the AP is -73m/2, which is option (c).