in a AP the first term is 8, nth term is 33 and sum to first n terms is 123. Find n and common difference

Step-by-step explanation:

First I want to set up some variables.

S = sum

n = final term

ax = xth term, so a1 is the first term and an is the last one

d = common difference.

There are two formulas to find the sum, if you don't get how they were gotten I'd be happy to explain.

S = (n/2) (a1+an) = (n/2) (2a1 + (n-1) d)

So, to find n we use the first one.

S = (n/2) (a1+an)

123 = (n/2) (8+33)

123 = (n/2) 41

3 = n/2

6 = n

Now we can find d with the other one

S = (n/2) (2a1 + (n-1) d)

123 = (6/2) (2*8 + (6-1) d)

123 = 3 (16+5d)

41 = 16+5d

25 = 5d

5 = d

so there are six terms and the common difference is 5.