Question

The sum of the first three numbers in an arithmetic progression is 18 . If the product of the first and the third term is 5 times the common difference, find the three numbers. ​

Answer 1

there is an alternative method also, taking the three numbers as: a-1, a, a+1.

then solving for a and substituiting value of a in the product of first and third term.

Hope it helps!

Answer 2

answer : 3, 6, 9 or, 15, 6, -3

Let a - d, a , a + d are first three numbers in an arithematic progression.

a/c to question,

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

or, 3a = 18

or, a = 6.......(1)

again, (a - d)(a + d) = 5d

or, a² - d² = 5d

from equation (1),

or, 6² - d² = 5d

or, 36 - d² = 5d

or, d² + 5d - 36 = 0

or, d² + 9d - 4d - 36 = 0

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

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

hence, d = 3 and -9

when d = 3 and a = 6

three numbers are ; 3, 6, 9

when d = -9 and a = 6

three numbers are ; 15, 6, -3