Question

the sum of the three numbers in AP is 9 if 4 is added to the third term then the resulting number are in GP find the numbers ​

Answer 1

Answer

1, 3, 5 or 9, 3, -3

$\rule{100}2$

Explanation

Let the three numbers in A.P. be (a - d), a and (a + d)

Then, Their sum as given

= a - d + a + a + d = 9

→ 3a = 9

→ a = 3

Now, if 4 is added to the third term, then the resulting series becomes a GP

Then, their common ratio will be equal

→ (a + d + 4)/a = a/(a - d)

→ a² = a² - d² +4a - 4d

→ d² + 4d - 4a = 0

Substituting the value of a,

→ d² + 4d - 4(3) = 0

→ d² + 4d - 12 = 0

→ d² + 6d - 2d - 12 = 0

→ d(d + 6) - 2(d + 6) = 0

→ (d - 2)(d + 6) = 0

→ d = 2 or d = -6

Then, for d = 2 the series would be

a - d, a, a + d

3 - 2, 3, 3 + 2

= 1, 3, 5

When 4 is added to 5, it becomes

1, 3, 9

which is in GP

Hence, the numbers 1, 3, 5 satisfies the condition

For d = -6,

3 - (-6), 3, 3 + (-6)

→ 3 + 6, 3, -3

→ 9, 3, -3

When 4 is added to -3, it becomes 1

and 9, 3, 1 is in GP

Hence the numbers 9, 3, -3 also satisfy the condition.

Answer 2

Given :----

• sum of three number = 9
• They are in AP also .
• if 4 is added to third number they are in GP .

To Find :----

• Value of all three numbers ..

Formula used :------

• if three terms are in GP , then (second term)² = (First term × third term)

Solution :------

Let the three numbers in AP be (a-d) , a and (a+d).

(where d is the common difference)

it is given that , their Sum is = 9

so,

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

→ 3a = 9

→ a = 3 = First term ..

So, our AP series will be (3-d) , 3 , and (3+d)

Adding 4 now in third term we get, = 3+d +4 = 7+d

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Now it is given that , if 4 is added to third term they will be in GP ,

so, now ,

(3-d), 3 and (7+d) Are in GP ..

Above Told formula for 3 GP terms we can conclude that ,

(3)² = (3-d)(7+d)

→ 9 = 21+3d-7d-d²

→ 9 = 21-4d - d²

→ d²+4d -12 = 0

solving the Quadratic Equation by splitting the middle term Now,

→ d² +6d - 2d - 12 = 0

→ d(d+6) -2(d+6) = 0

→(d+6)(d-2) = 0

d = (-6) or 2 .

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If d = (-6)

our three AP terms will be :----

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

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

So our series will be :--- 9 , 3 and (-3)

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If d = 2

(3-d) = 3-2 = 1

(3+d) = 3+2 = 5

so, our series will be :--- 1 , 3 & 5 ..

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Extra Brainly knowledge :--

when three numbers a,b and c are in AP ,

than ,

2b = a +c

or,

b = (a+c)/2

is the relation between them ..

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(Hope this Helps you)