the terms of an arithmetic progression are given by the formula,

a_n = a_1 + (n - 1)d

where a_n is the nth term, a_1 is the first term, n is the current number of the term and d is the common difference.

we are told the 20th term is 10

=> a_20 = a_1 + (20 - 1)d = a_1 + 19d = 10

we are told the 50th term is 70

=> a_50 = a_1 + (50 - 1)d = a_1 + 49d = 70

so to find the first term (a_1) and the common difference (d) we must solve the system:

a_1 + 19d = 10 ....................(1)

a_1 + 49d = 70 ....................(2)

=> 30d = 60 ........................(1) - (2)

=> d = 2

but a_1 + 19d = 10

=> a_1 + 19(2) = 10

=> a_1 = -28

so the first term is -28 and the common difference is 2

From above, we see that the formula for our progression will be:(ii) Show that the su of the first 29 terms is zero.

a_n = -28 + (n - 1)2

so a_n = -30 + 2n for n = 1,2,3,4,5...

what is the 29th term?

a_29 = -30 + 2(29) = 28

Now, the sum of the first n terms of an arithmetic progression is given by the formula,

S_n = n(a_1 + a_n)/2

so the sum of the first 29 terms is:

S_29 = 29(-28 + 28)/2 = 0