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.