Question

the sum of the 4th and 8th term of an ap is 20 and the sum of the 6th and 10th term is 36. Find the three terms of the AP​

Answer 1

ANSWER

Given

The sum of the 4th and the 8th term of an AP is 20.

$\implies a+3d+a+7d=20$

$\implies 2a+10d=20$

Dividing by 2, we get,

$\implies a+5d=10--(1)$

The sum of the 6th term and the 10th term is 36.

$\implies a+5d+a+9d=36$

$\implies 2a+14d=36$

Dividing by 2, we get,

$\implies a+7d=18--(2)$

Subtracting (2) from (1), we get,

$\implies a+7d-(a+5d)=18-10\\\implies a+7d-a-5d=8\\\implies 2d=8\\\implies \underline{{d = 4}}}}$

Substituting d=4 in equation (1), we get,

$\implies a + 5(4) = 10\\\implies a+20=10\\\implies a = 10-20\\\implies \underline{\underline{a=-10}}}$

=> a₁ = -10

=> a₂ = -10 + 4 = -6

=> a₃ = -10 + 2(4) = -10 + 8 = -2

$\boxed{\boxed{\bold{The \ 3 \ terms \ of \ the \ AP \ are \ -10, \ -6 \ and \ -2.}}}}}}$

                                                               

Answer 2

EXPLANATION.

Sum of 4th and 8th term of an ap = 20.

Sum of the 6th and 10th term of an ap = 36.

As we know that

General term of an ap.

⇒ Tₙ = a + (n - 1)d.

Using this formula in the equation, we get.

Sum of 4th and 8th term of an ap = 20.

⇒ T₄ + T₈ = 20.

⇒ a + (4 - 1)d + a + (8 - 1)d = 20.

⇒ a + 3d + a + 7d = 20.

⇒ 2a + 10d = 20.

⇒ a + 5d = 10. - - - - - (1).

Sum of the 6th and 10th term of an ap = 36.

⇒ T₆ + T₁₀ = 36.

⇒ a + (6 - 1)d + a + (10 - 1)d = 36.

⇒ a + 5d + a + 9d = 36.

⇒ 2a + 14d = 36.

⇒ a + 7d = 18. - - - - - (2).

From equation (1) and (2), we get.

Subtracting equation (1) and (2), we get.

⇒ a + 5d = 10. - - - - - (1).

⇒ a + 7d = 18. - - - - - (2).

⇒ -  -          -

We get,

⇒ - 2d = - 8.

⇒ d = 4.

Put the values of d = 4 in equation (1), we get.

⇒ a + 5d = 10.

⇒ a + 5(4) = 10.

⇒ a + 20 = 10.

⇒ a = 10 - 20.

⇒ a = - 10.

First term = a = - 10.

Common difference = d = b - a = 4.

As we know that,

Arithmetic series can be written as,

⇒ a, a + d, a + 2d, . . . . .

⇒ (-10), (-10 + 4), (-10 + 2(4)), . . . . .

⇒ - 10, - 6, - 2, . . . . . .

Three terms of the ap = - 10, - 6, - 2.