Question

The sum of 6th and 8th term of an Ap is 142 and if the 4th term is 49. Find a and d​

Answer 1

Answer:

The values of a and d are 27 & 22 / 3.

Step-by-step explanation:

We have given that,

Sum of 6ᵗʰ and 8ᵗʰ term of an A.P. is 142.

We know that,

tₙ = a + ( n - 1 ) * d - - [ Formula ]

∴ t₆ = a + ( 6 - 1 ) * d

⇒ t₆ = a + 5d - - ( 1 )

Now,

t₈ = a + ( 8 - 1 ) * d

⇒ t₈ = a + 7d - - ( 2 )

From the first condition,

t₆ + t₈ = 142

⇒ a + 5d + a + 7d = 142 - - [ From ( 1 ) & ( 2 ) ]

⇒ a + a + 5d + 7d = 142

⇒ 2a + 12d = 142

⇒ a + 6d = 71 - - ( 3 ) [ Dividing by 2 ]

Now, by using the formula tₙ = a + ( n - 1 ) * d,

t₄ = a + ( 4 - 1 ) * d

⇒ t₄ = a + 3d

⇒ 49 = a + 3d

⇒ a + 3d = 49 - - ( 4 )

By subtracting equation ( 4 ) from equation ( 3 ), we get,

⇒ a + 6d - ( a + 3d ) = 71 - 49

⇒ a + 6d - a - 3d = 22

⇒ 3d = 22

⇒ d = 22 / 3

By substituting d = 22 / 3 in equation ( 4 ), we get,

a + 3d = 49 - - ( 4 )

⇒ a + 3 * ( 22 / 3 ) = 49

⇒ a + 22 × 3 / 3 = 49

⇒ a + 22 × 1 = 49

⇒ a + 22 = 49

⇒ a = 49 - 22

⇒ a = 27

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Additional Information:

1. Arithmetic Progression:

1. In a sequence, if the common difference between two consecutive terms is constant, then the sequence is called as Arithmetic Progression ( AP ).

2. nᵗʰ term of an AP:

The number of a term in the given AP is called nᵗʰ term of an AP.

3. Formula for nᵗʰ term of an AP:

• tₙ = a + ( n - 1 ) * d

4. The sum of the first n terms of an AP:

The addition of either all the terms of a particular terms is called as sum of first n terms of AP.

5. Formula for sum of the first n terms of A. P. :

• Sₙ = n / 2 [ 2a + ( n - 1 ) * d ]

Answer 2

★Given:

• 4 th term (t4) = 49

• t6+t8= 142

★To Find:

• a=?

• d=?

★Solution:

Let a be the first term and d be the common difference.

tn= a+(n-1) d...... (formula)

t6 = a+(6-1) d

: . t6=a+5d.................. (1)

and t8=a+7d.............. (2)

Adding equations (1) and (2) we get,

t6+t8=a+5d+a+7d

: . 142=2a+12d

: . 71=a+6d..

: . a+6d=71.............. (3)

Now,

t4=a+3d

: . 49=a+3d...........(4)

Subtracting equation (4) from (3)

a+6d=71

-

a+3d=49

___________

3d=22

: . d=22/3

Substituting the value of d in equation (4)

a+3(22/3) =49............. (4)

: . a+22=49.

: . a=49-22

: . a=27

THEREFORE,

THE VALUE OF D IS 22/3 AND A IS 27...