Question

The sum of first four term of an ap is 40 the ratio of the products of first and fourth term to second and third term is 2:3 find these terms

Answer 1

Answer:

a+d,a-d,a+3d,a-3d are terms

Step-by-step explanation:

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Answer 2

The A.P. is 16, 12, 8, 4, ..... , or the A.P. is 4, 8, 12, 16, ......... .

Step-by-step explanation:

Let the first four unknown terms of the A.P. are

a, (a + d), (a + 2d) and (a + 3d).

Now, the first four terms of the A.P. has sum 40.

So, a + (a + d) + (a + 2d) + (a + 3d) = 40

⇒ 4a + 6d = 40

⇒ 2a + 3d = 20 ........... (1)

Now, the ratio of the products of first and fourth term to second and third term is 2:3.

Therefore,

$\frac{a(a + 3d)}{(a + d)(a + 2d)} = \frac{2}{3}$

⇒ 3a² + 9ad = 2(a² + 3ad + 2d²)

⇒ 3a² + 9ad = 2a² + 6ad + 4d²

⇒ a² + 3ad - 4d² = 0

⇒ a² + 4ad - ad - 4d² = 0

⇒ (a + 4d)(a - d) = 0

So, a = - 4d or a = d

Now, from equation (1) we get

when, a = - 4d, then 2(- 4d) + 3d = 20

⇒ - 5d = 20

⇒ d = - 4

So, a = - 4(- 4) = 16

Hence, the A.P. is 16, 12, 8, 4, .....

Now, if a = d, then 5d = 20

⇒ d = 4

So, a = d = 4

Hence, the A.P. is 4, 8, 12, 16, ........ (Answer)