How to find the n sequence in an arithmetic sequence with two given terms?

Question

 a5 = 15, a8 = 30

Answer 1

In arithmetic sequence n^th  term = a + (n - 1)d

Where a is first term and d is common difference.

5^th  term = a + 4d

8^th  term = a + 7d

a + 4d = 15 ---> (1)

a + 7d = 30 ---> (2)

Solve the equations (1) and (2).

From (1), a = 15 - 4d

Substitute the value of a in equation (2).

15 - 4d + 7d = 30

15 + 3d = 30

3d = 30 - 15

3d = 15

d = 15/3

d = 5

Substitute the value of d in equation (1).

a + 4(5) = 15

a + 20 = 15

a = - 20 + 15

a = - 5

In this case there no possibility for find n without giving additional data.

Find the n^th term

Formula for the n^th  term = a + (n - 1)d

= - 5 + (n - 1)5

= - 5 + 5n - 5

n^th term is  5n - 10.

Find the sum of the sequence

Sn = n/2 [ 2a + (n - 1)d]

= n/2 [ 2(- 5) + (n - 1)5]

= n/2 [ - 10 + 5n - 5]

= n/2 [ 5n - 15]

Sum of the sequence = (5n²/2) - (15n/2).