Question

the sum of three numner in A.P is 15, when 1,4 and 19 are added to these numbers respectively, they are found to ne in G.P find them.​

Answer 1

$\huge\red{\underline{\tt{\overbrace{\underbrace{Hello } }}}}$

★ qUESTIOn ★

the sum of three numner in A.P is 15, when 1,4 and 19 are added to these numbers respectively, they are found to ne in G.P find them.

† sOLUTIOn †

→ Let the numbers be = a-d, a+d

then,

→ a-d+a+a+d=15

→ 3a=15

→ a=5

When,

→ 1 , 4 and 19 are added respectively

→ (a-d+1) ,(a+4),(a+d+19) are in G.P

→ (6-d),(9),(24+d) are in G.P

$\implies (6-d) (24+d) = 9^2\\ \implies 144-18d-d^3=81\\ \implies d^3+18d-63=0\\ \implies d^2+21d-3d-63=0\\ \implies d^2+21d-3d-63=0\\ \implies d(d+21)-3(d+21)=0\\ \implies(d+21)(d-3)=0\\ \implies d=-21 \: or\:3$

HENCE,

→ Let d = -21 , then the number are = 5 -(-21) , 5,

(5-21)

or 26 , 5 and - 16 Answer

→ let d = 3 then the number are (5-3) , 5 and (5+3)

or 2 ,5 and 8 Answer.

$\rule{230}2$

Answer 2

AnswEr :

Given that sum of three numbers in A.P.is 15. When 1, 4 & 19 are added to each respectively, they are found to be in G.P.

We have to find the three numbers.

Let the three numbers be a, a + d and a + 2d.

According to question :

→ a + a + d + a + 2d = 15

→ 3a + 3d = 15

→ 3(a + d) = 15

→ a + d = 15/3

→ a + d = 5

→ a = 5 - d ...(i)

Now let's find the three numbers after adding 1, 4 & 19.

Now three numbers are :

→ First number = a + 1 = 5 - d + 1 = 6 - d

→ Second number = a + d + 4 = 5 - d + d + 4 = 9

→ Third number = a + 2d + 19 = 5 - d + 2d + 19 = d + 24

Now let the three numbers be a, ar & ar² [As they are in G.P.]

From these three terms we get the rules of G.P. :-

→ a × ar² = a²r² [1st term × 3rd term]

→ (ar)² = a²r² [2nd term²]

Thus we can conclude that in a G.P. :-

→ 1st term × 3rd term = (2nd term)²

Substituting values,

→ (6 - d) × (d + 24) = 9²

→ 6d + 144 - d² - 24d = 81

→ -d² - 18d + 144 - 81 = 0

→ -(d² + 18d - 63) = 0

→ d² + 18d - 63 = 0

→ d² + 21d - 3d - 63 = 0

→ d(d + 21) - 3(d + 21) = 0

→ (d - 3)(d + 21) = 0

→ d = 3 or, d = -21

Now taking d = 3 , putting in (i) :

→ a = 5 - 3

→ a = 2

Taking d = -21,

→ a = 5 - (-21)

→ a = 5 + 21

→ a = 26

Now the three numbers in A.P. :-

→ First number = a = 2

→ Second number = a + d = 2 + 3 = 5

→ Third number = a + 2d = 2 + 2(3) = 2 + 6 = 8

∴ Three numbers : 2, 5 and 8 .

Again taking a = 26 and d = -21,

Three numbers are :

→ First number = a = 26

→ Second number = a + d = 26 + (-21) = 26 - 21 = 5

→ Third number = a + 2d = 26 + 2(-21) = 26 - 42 = -16

∴ Three numbers are : 26, 5 and -16 .

Therefore,

The three numbers in A.P. are : 2,5, 8 or, 26, 5, - 16 .