The 3rd,4th and 5th term of an Arithmetic progression(series)are (4+x),2x and (8-x).find the value of sum of x and the sum of the first 24terms of the series.

Answer 1

To find the value of "x," we can use the formula for the nth term of an arithmetic progression:

aₙ = a₁ + (n-1)d

where aₙ is the nth term, a₁ is the first term, n is the term number, and d is the common difference.

Given that the 3rd, 4th, and 5th terms are (4+x), 2x, and (8-x), respectively, we can set up the following equations:

(4+x) = a₁ + 2d ... (equation 1)
2x = a₁ + 3d ... (equation 2)
(8-x) = a₁ + 4d ... (equation 3)

To solve these equations, we can subtract equation 1 from equation 2, and equation 2 from equation 3:

(2x - (4+x)) = ((a₁ + 3d) - (a₁ + 2d))
2x - 4 - x = a₁ + 3d - a₁ - 2d
x - 4 = d

((8-x) - 2x) = ((a₁ + 4d) - (a₁ + 3d))
8 - 3x = a₁ + 4d - a₁ - 3d
8 - 3x = d

Setting the equations for d equal to each other:

x - 4 = 8 - 3x
4x = 12
x = 3

Therefore, the value of x is 3.

To find the sum of the first 24 terms of the series, we can use the formula for the sum of an arithmetic series:

Sₙ = n/2 * (a₁ + aₙ)

where Sₙ is the sum of the first n terms.

Given that n = 24, we can substitute the values we know to find the sum:

S₂₄ = 24/2 * (a₁ + a₂₄)

We can find a₁ and a₂₄ by substituting x = 3 into the equations we derived earlier:

a₁ = (4 + x) = (4 + 3) = 7
a₂₄ = (a₁ + 23d) = (7 + 23(d = x - 4)) = (7 + 23(3 - 4)) = 7 - 23 = -16

Substituting these values into the equation for S₂₄:

S₂₄ = 24/2 * (7 + (-16))
S₂₄ = 12 * (-9)
S₂₄ = -108

Therefore, the sum of the first 24 terms of the series is -108.