Question

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

Answer 1

Answer:

The first term is -3.

Step-by-step 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

$a_5 + a_6 = 30$

$5 + 2d + 5 + 3d = 30$

$5d + 10 = 30$

$5d = 20$

$d = \frac{20}{5} = 4$

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.