Question

The sum of the first five terms of an arithmetic sequence is 20 and the sixth term is 25. What is the first term of the sequence and the common difference?

Answer 1

S5 = 20

A6 = 25

A1=

d=

The general formula for arithmetic sequence is:

An = A1 + (n - 1)d

25 = A1 + 6d - d (1)

And the formula for arithmetic sum:

Sn = n(A1 + An)/2

20 = 5(A1 + 25 - d)/2

8 = A1 + 25 - d (2)

We have a system of two equations with two variables.

25 = A1 + 5d

8 = A1 + 25 - d

For both equations, we can solve for d and then equal both equations as follows:

5 - A1/5 = d (1)

A1 + 17 = d (2)

A1 + 17 = 5 - A1/5

A1 = -10.

Now we just replace the value of A1 in equation 1:

d = 7