Question

Find the first term and the common difference of the arithmetic sequence whose 6th term is 30 and 12th term

is 54.

Answer 1

Given :-

6th term = 30

12th term = 54

To Find :-

First term

Common difference

Solution :-

We know that

`\sf \: a_(n) = a + (n - 1)d`

For 6th term

`\sf \: 30 = a + (6 - 1)d`

`\sf \: 30 = a + 5d`

For 12th term

`\sf \: a_(n) = a + (n - 1)d`

`\sf \: 54 = a + (12 - 1)d`

`\sf \: 54 = a + 11d`

On subtracting both

`\sf \: 54 - 30 = a + 11d - (a + 5d)`

`\sf \: 54 - 30 = a + 11d - a - 5d`

`\sf \: 24 = 6d`

`\sf \: (24)/(6) = d`

`\sf \: 4 = d`

Now

Using 2

`\sf \: 54 = a + 11d`

`\sf \: 54 = a + 11(4)`

`\sf \: 54 = a + 44`

`\sf \: 54 - 44 = a`

`\sf \: 10 = a`

Answer 2

• 10 and 4

Explanation:

The first term is a, common difference is d.

The 6th term:

• a + 5d = 30

The 12th term:

• a + 11d = 54

Solve the system by elimination:

• 11d - 5d = 54 - 30
• 6d = 24
• d = 4

Find a:

• a + 5*4 = 30
• a + 20 = 30
• a = 10