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The sum of three numbers in AP is 3 and their product is –35. Find the numbers.

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Let numbers a, b and c are in AP. 

∴ 2b = a + c. … (1) 

Given that sum of these numbers is 3. 

∴ a + b + c = 3 

⇒ 2b + b = 3 (From equation (1), a + c = 2b) 

⇒ 3b = 3 

⇒ b = 1. 

Now putting b = 1, in equation (1), we get 

a + c = 2. … (2) 

Given that the product of these numbers is –35

∴ abc = –35 

⇒ ac = –35. (∵ b =1) 

Now ( a − c )2 = ( a + c )2 − 4ac = 22 – (4 × – 35) = 4 + 140 = 144. 

(∵ a + c = 2, ac = –35) 

⇒ a – c = \(\sqrt{144}\) = 12 

⇒ a – c = 12. … (3) 

Now, adding equations (2) and (3), we get 

(a + c)+(a – c) = 2 +12 = 14 

⇒ 2a = 14 

⇒ a = \(\frac{14}2\) = 7. 

Now, putting a = 7 in equation (2), we get 

7 + c = 2 ⇒ c = 2 –7 = –5. 

∴ a = 7, b = 1 and c = –5.

Hence, the numbers are 7, 1 and –5.

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