Question

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Answer 1

Let the numbers be ( a - d), a ( a+d ).

( a - d) + a + ( a +d ) = 18

3a = 18

a = 6

First term = a - d = 6 - d

Third term = a+d = 6 + d

(6 - d) ( 6 +d) = 5d

36 + 6d - 6d - d^2 = 5d

d^2 + 5d - 36 = 0

d^2 + 9d - 4d - 36 = 0

d( d +9 ) - 4( d +9 ) = 0

( d - 4) ( d + 9 ) = 0

d = 4, - 9

Common difference = 4

Numbers = ( 6 - d) = 2

a = 6

( 6 +d) = 10

A. P. = 2, 6, 10...

For CD. = - 9

( a-d) = 6 + 9 = 15

( a) = 6

( a +d ) = 6 - 9 = - 3

AP. = 15 , 6, - 3

Answer 2

Answer:

2,6,10 (or) 15,6,-3

Step-by-step explanation:

Let the three terms be a - d, a, a + d.

Given, Sum of first three numbers is 18.

⇒ a - d + a + a + d = 18

⇒ 3a = 18

⇒ a = 6

Given, Product of first and the third term is 5 times the common difference

⇒ (6 - d)(6 + d) = 5d

⇒ 36 - d² = 5d

⇒ -d² - 5d + 36 = 0

⇒ d² + 5d - 36 = 0

⇒ d² + 9d - 4d - 36 = 0

⇒ d(d + 9) - 4(d + 9) = 0

⇒ (d - 4)(d + 9) = 0

⇒ d = 4, -9

When a = 6, d = 4:

a - d = 2

a = 6

a + d = 10

When a = 6, d = -9:

a - d = 15

a = 6

a + d = -3

Therefore, the three numbers are 2,6,10 (or) 15,6,-3

Hope it helps!