The 8th term of ap is 5time the 3rd term, while the 7th term is greater than the 4th term, write the first 5 term of the ap

YES

The third term of an a.p. is 9 while the 11th terms is -7 find the first five terms of the a.p

To write the first 5 terms of the arithmetic progression (AP), we need to find the common difference (d) of the AP.

We are given that the 8th term is 5 times the 3rd term. Let's denote the 3rd term as "a" and the common difference as "d."

The 8th term can be written as "a + 7d" (since the 8th term is a + 7d positions away from the 1st term), and we are given that it is 5 times the 3rd term, which can be written as "a + 2d" (since the 3rd term is a + 2d positions away from the 1st term).

Setting up the equation: a + 7d = 5(a + 2d)

Expanding: a + 7d = 5a + 10d

Rearranging: 7d - 10d = 5a - a

Simplifying: -3d = 4a

Dividing both sides by 4: -3d/4 = a

Now we have found the 3rd term (a) in terms of the common difference (d).

Next, we are given that the 7th term is greater than the 4th term. The 7th term can be written as "a + 6d" (since the 7th term is a + 6d positions away from the 1st term), and the 4th term can be written as "a + 3d" (since the 4th term is a + 3d positions away from the 1st term).

Setting up the inequality: a + 6d > a + 3d

Simplifying: 6d > 3d

Since the common difference (d) is positive, we can divide both sides of the inequality by 3 to get the result: d > 0

Now that we know the relationship between the common difference and the third term, and also the constraint on the common difference, we can proceed to write the first 5 terms of the AP.

The first term (a) is -3d/4, and the common difference (d) is greater than 0. Therefore, the five terms can be written as:

1st term: -3d/4
2nd term: -3d/4 + d
3rd term: -3d/4 + 2d
4th term: -3d/4 + 3d
5th term: -3d/4 + 4d

Simplifying each term, we get:

1st term: -3d/4
2nd term: d/4
3rd term: 5d/4
4th term: 7d/4
5th term: 9d/4

Therefore, the first 5 terms of the arithmetic progression are -3d/4, d/4, 5d/4, 7d/4, and 9d/4.