Question

the sum of the three consecutive terms of an ap is 12 and product is 48 find all terms of progression

Answer 1

In AS( arithmetic sequence ),

First term = a

Common difference = d

Second term = a₂ = a + d

Third term = a₃ = a + 2d

xth term = aₓ = a + ( x - 1 )d

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Let the consecutive terms of AS be ( a - d ) , a , ( a + d ).

In the question it is given that the sum of the consecutive terms of that AS is 12.

⇒ ( a - d ) + a + ( a + d ) = 12

⇒ a - d + a + a + d = 12

⇒ a + a + a - d + d = 12

⇒ 3a = 12

⇒ a = 12 / 3

⇒ a = 4

Also given that the product of the same terms in the AS is 48.

⇒ ( a - d ) ( a ) ( a + d ) = 48

⇒ ( a - d ) ( a + d ) a = 48

⇒ ( a^2 - d^2 ) a = 48

Substituting the value of a :

⇒ ( 4^2 - d^2 ) 4 = 48

⇒ 16 - d^2 = 48 / 4

⇒ 16 - d^2 = 12

⇒ 16 - 12 = d^2

⇒ 4 = d^2

⇒ $\pm 2 = d$

⇒ 2 or - 2 = d

Now, there are two values of d or common difference, substituting both the values of d in terms of AP one by one.

If we take the value of d equal to 2. Arithmetic progressions are :

a - d = 4 - 2 = 2

a = 4

a + d = 4 + 2 =  6

If we take the value of d equal to - 2. Arithmetic progressions are :

a - d = 4 - ( - 2 ) = 4 + 2 = 6

a = 4

a + d = 4 + ( - 2 ) = 4 - 2 = 2

Answer 2

Here is your solution

Given :-

In  arithmetic sequence ,

Let,First term = a

Common difference = d

Second term= a₂ = a + d

Third term = a₃ = a + 2d

The consecutive terms of AS be ( a - d ) , a , ( a + d ).

sum of the consecutive terms of that AS is 12.

A/q

=>( a - d ) + a + ( a + d ) = 12

=>a - d + a + a + d = 12

=>a + a + a - d + d = 12

=>3a = 12

=>a=12 / 3

=>a=4........ (i)

Now

given that the product of the same terms in the AS is 48.

A/q

=>( a - d ) ( a ) ( a + d ) = 48

=>( a - d ) ( a + d ) a = 48

=>( a^2 - d^2 ) a = 48..... (ii)

Putting the value of a  in equation (ii)

=>( 4^2 - d^2 ) 4 = 48

=>16 - d^2 = 48 / 4

=>16 - d^2 = 12

=>16 - 12 = d^2

=>4 = d^2

=>d=+ -2

Now,

We have already find  two values of d or common difference, putting  both the values of d in terms of AP .

Firstly we take the value of d = 2.

Arithmetic progressions are :-

a - d = 4 - 2 = 2

a = 4

a + d = 4 + 2 =  6

Ap are 2,4,6

Then If we take the value of d = (- 2)

Arithmetic progressions are :-

a - d = 4 - ( - 2 ) = 4 + 2 = 6

a = 4

a + d = 4 + ( - 2 ) = 4 - 2 = 2

Ap are 6,4,2

Hope it helps you