To find the first term of the arithmetic progression (AP), we can use the formula:
an = a1 + (n-1)d
where an is the nth term, a1 is the first term, n is the number of terms, and d is the common difference.
Given that the 2nd term is 11, we can substitute these values into the formula:
11 = a1 + (2-1)d
11 = a1 + d
Similarly, given that the 6th term is -17, we can substitute these values into the formula:
-17 = a1 + (6-1)d
-17 = a1 + 5d
Now we have a system of two equations:
11 = a1 + d
-17 = a1 + 5d
We can solve this system of equations to find the values of a1 and d.
Subtracting the first equation from the second equation, we get:
-17 - 11 = (a1 + 5d) - (a1 + d)
-28 = 4d
Dividing both sides by 4, we get:
d = -7
Substituting this value of d into the first equation, we can solve for a1:
11 = a1 + (-7)
11 = a1 - 7
a1 = 11 + 7
a1 = 18
So, the first term of the AP is 18.
b) The common difference is -7.
c) To find the sum of the first 50 terms, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(2a1 + (n-1)d)
Substituting the values, we get:
S50 = (50/2)(2(18) + (50-1)(-7))
S50 = 25(36 + 49(-7))
S50 = 25(36 - 343)
S50 = 25(-307)
S50 = -7675
So, the sum of the first 50 terms is -7675.