Find the four numbers in A.P., whose sum is 50 and in which the greate

1.	Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
Answer» Let’s consider the four terms of the A.P. to be (a – 3d), (a – d), (a + d) and (a + 3d). From the question, Sum of these terms = 50 ⇒ (a – 3d) + (a – d) + (a + d) + (a + 3d) = 50 ⇒ a – 3d + a – d + a + d + a – 3d= 50 ⇒ 4a = 50 ⇒ a = 50/4 = 25/2 And, also given that the greatest number = 4 x least number ⇒ a + 3d = 4 (a – 3d) ⇒ a + 3d = 4a – 12d ⇒ 4a – a = 3d + 12d ⇒3a = 15d ⇒a = 5d Using the value of a in the above equation, we have ⇒25/2 = 5d ⇒ d = 5/2 So, the terms will be: (a – 3d) = (25/2 – 3(5/2)), (a – d) = (25/2 – 5/2), (25/2 + 5/2) and (25/2 + 3(5/2)). ⇒ 5, 10, 15, 20

Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.

Answer»

Let’s consider the four terms of the A.P. to be (a – 3d), (a – d), (a + d) and (a + 3d).

From the question,

Sum of these terms = 50

⇒ (a – 3d) + (a – d) + (a + d) + (a + 3d) = 50

⇒ a – 3d + a – d + a + d + a – 3d= 50

⇒ 4a = 50

⇒ a = 50/4 = 25/2

And, also given that the greatest number = 4 x least number

⇒ a + 3d = 4 (a – 3d)

⇒ a + 3d = 4a – 12d

⇒ 4a – a = 3d + 12d

⇒3a = 15d

⇒a = 5d

Using the value of a in the above equation, we have

⇒25/2 = 5d

⇒ d = 5/2

So, the terms will be:

(a – 3d) = (25/2 – 3(5/2)), (a – d) = (25/2 – 5/2), (25/2 + 5/2) and (25/2 + 3(5/2)).

⇒ 5, 10, 15, 20