Question

the sum of the first 3 terms in an ap is 18.if the product of the first and the third term is 5 times the common different,find the three number.

Answer 1

I hope my answer helps u

Answer 2

Answer:

3 terms are 15 , 6 , -3 or 2 , 6 , 10.

Step-by-step explanation:

Given the sum of first three terms of an AP = 18

Product of 1st and 3rd term is 5 times common difference

we have to find the three numbers.

Let  the first three terms of an AP are (a - d) , a , (a + d)

The first term of AP = a - d

And common difference = 2nd term - 1st term = a - ( a - d )

                                         = a - a + d = d

Sum of first three term of AP = 18

⇒ ( a - d ) + ( a ) + ( a + d ) = 18

⇒  a - d + a + a + d = 18

⇒  3a = 18 ⇒ a = 6

Now, Product of 1st and 3rd term is 5 times common difference

⇒ $( a - d ) \times ( a + d ) = 5 \times d$

⇒ $a^2 - d^2 = 5d$     ( using identity, ( x - y )( x + y ) = x² - y² )

⇒ $6^2 - d^2 = 5d$

⇒ $d^2 + 5d - 36 = 0$

⇒ $d^2 + 9d -4d - 36 = 0$

⇒ $d(d + 9) - 4 ( d + 9 ) = 0$

⇒ $(d + 9 ) ( d - 4 ) = 0$

⇒ $d = - 9 , 4$

when d = -9

⇒ $a - d = 6 - (-9) = 15 , a = 6 , a + d = 6 + (-9) = -3$

when d = 4

⇒ $a - d = 6 - (4) = 2 , a = 6 , a + d = 6 + (4) = 10$

Therefore, 3 terms are 15 , 6 , -3 or 2 , 6 , 10.