Question

the sum of the first three numbers in an ap is 18 if the product of the first and the third term is 5 times the common difference find the three numbers​

Answer 1

Answer ;

the three numbers are a+d, a, a-d

question is sum of first three numbers is 18

• so a+d+a+a-d=18

• d and -d will be cancelled
• so a+a+a=18
• 3a=18
• a=18/3
• a=6    
• product of frist and third term is 5times common difference
• so (a+d)(a-d)=5d
• so a^2-d^2=5d here a=6
• 6^2-d^2=5d
• 36-d^2=5d
• -d^2/d=5-36
• -d= -31
• d=31
• so the numbers are a+d =6=31=36
• a-d=6-31= -25
• a=6

Answer 2

The three numbers are "2, 6 and 10 or 10, 6, 2".

Step-by-step explanation:

Let a - d, a and a + d are the three terms of an AP.

To find, the three numbers = ?

According to question,

$a - d+a+a+d=18$

⇒ $3a=18$

⇒ a = 6

Also,

$(a - d)(a+d)=5d$

⇒ $a^{2}-d^{2}=5d$

⇒ $6^{2}-d^{2}=5d$

⇒$d^{2} +5d-36=0$

⇒$d^{2} +9d-4d-36=0$

⇒$(d-4)(d+9)=0$

∴ d = 4 or, - 9

The three numbers are:

6 - 4, 6 and 6 + 4  or 2, 6 and 10 or 10, 6, 2.

Hence, the three numbers are 2, 6 and 10 or 10, 6, 2.