Question

Find four numbers in ap whose sum is 20 and the sum of whose squares is 180

Answer 1

above is the required answer of the question

Answer 2

The four numbers are - 1 , 3 , 7 , 11

Given :

• Four numbers are in AP

• Sum of four numbers is 20

• The sum of whose squares is 180

To find :

Four numbers

Solution :

Step 1 of 4 :

Assume four numbers are in AP

Here it is given that four numbers are in AP

Let four numbers are a - 3d , a - d , a + d , a + 3d .

Step 2 of 4 :

Use the first condition

Here it is given that sum of four numbers is 20

(a - 3d) + (a - d) + (a + d) + (a + 3d) = 20

⇒ 4a = 20

⇒ a = 5

Step 3 of 4 :

Use second condition

(a - 3d)² + (a - d)² + (a + d)² + (a + 3d)² = 180

⇒ a² - 6ad + 9d² + a² - 2ad + d² + a² + 2ad + d² + a² + 6ad + 9d² = 180

⇒ 4a² + 20d² = 180

⇒ a² + 5d² = 45

⇒ 5² + 5d² = 45

⇒ 25 + 5d² = 45

⇒ 5d² = 20

⇒ d² = 4

⇒ d = ± 2

Step 4 of 4 :

Find the four numbers

Taking d = 2

Four numbers are - 1 , 3 , 7 , 11

Taking d = - 2

Four numbers are 11 , 7 , 3 , - 1

Hence four numbers are - 1 , 3 , 7 , 11

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