Question

The sum of the fifth and sixth term of an AP is 30.If the third term is 5.Find the first term.

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Answer 1

The first term is -3.

Explanation:

Arithmetic sequence:

In an arithmetic sequence, the difference between two consecutive terms is always the same, and it is called common difference d.

The nth term is given by:

`a_n = a_1 + (n-1)d`

In which
`a_1` is the first term.

If we have the mth term instead of the first, we have:

`a_n = a_m + (n-m)d`

We have the third term, and need the fifth and the sixth:

So we write both the fifth and the sixth in function of the third, then:

`a_5 = a_3 + 2d = 5 + 2d`

`a_6 = a_3 + 3d = 5 + 3d`

The sum of the fifth and sixth term of an AP is 30

We use this to find
`d`. So

`a_5 + a_6 = 30`

`5 + 2d + 5 + 3d = 30`

`5d + 10 = 30`

`5d = 20`

`d = (20)/(5) = 4`

First term:

The third term is:

`a_3 = a_1 + 2d`

Since
`a_3 = 5, d = 4`

`a_1 = a_3 - 2d = 5 - 8 = -3`

The first term is -3.