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.
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.
To find the value of x, we can use the fact that an arithmetic progression has a common difference between each term.
We know that the 3rd term of the progression is (4+x), the 4th term is 2x, and the 5th term is (8-x).
The difference between the 3rd and 4th terms is the same as the difference between the 4th and 5th terms. Therefore, we can set up the following equation:
(4 + x) - 2x = 2x - (8 - x)
Simplifying this equation, we get:
4 + x - 2x = 2x - 8 + x
4 - x = 3x - 8
Combining like terms, we have:
4 + 8 = 3x + x
12 = 4x
Dividing both sides by 4, we find that x = 3.
Now, 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)(2a + (n-1)d)
Here, n represents the number of terms, a represents the first term, and d represents the common difference.
For our series, the first term (a) is (4 + x) = 7, the number of terms (n) is 24, and the common difference (d) is 2x - (4 + x) = 2(3) - (4 + 3) = 6 - 7 = -1.
Substituting these values into the formula, we have:
S24 = (24/2)(2(7) + (24 - 1)(-1))
= 12(14 + 23(-1))
= 12(14 - 23)
= 12(-9)
= -108
Therefore, the sum of x is 3 and the sum of the first 24 terms of the series is -108.
To find the value of x and the sum of the first 24 terms of the series, we can use the formula for the nth term of an arithmetic progression and the formula for the sum of the first n terms of an arithmetic progression.
The formula for the nth term of an arithmetic progression is:
a_n = a_1 + (n-1)d
where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference.
In this case, we are given the 3rd, 4th, and 5th terms:
a_3 = 4 + x
a_4 = 2x
a_5 = 8 - x
Let's use this information to find the common difference (d) and the value of x.
Since the common difference between consecutive terms in an arithmetic progression is constant, we can find d by subtracting the 3rd term from the 4th term:
d = a_4 - a_3 = 2x - (4 + x) = 2x - 4 - x = x - 4
Similarly, we can find d by subtracting the 4th term from the 5th term:
d = a_5 - a_4 = (8 - x) - 2x = 8 - 3x
Since both expressions for d must be equal, we can equate them and solve for x:
x - 4 = 8 - 3x
4x = 12
x = 3
Now that we have the value of x, we can find the common difference (d):
d = x - 4 = 3 - 4 = -1
Now let's move on to finding the sum of the first 24 terms of the series using the formula for the sum of an arithmetic progression:
S_n = (n/2)(a_1 + a_n)
where S_n is the sum of the first n terms, a_1 is the first term, a_n is the nth term, and n is the number of terms.
We are asked to find the sum of the first 24 terms, so n = 24. We already know the value of the first term (a_1 = 4 + x = 4 + 3 = 7), and we can now find the value of the 24th term (a_24) using the formula for the nth term:
a_24 = a_1 + (24-1)d
a_24 = 7 + 23(-1)
a_24 = 7 - 23
a_24 = -16
Now we can substitute these values into the formula for the sum of the first 24 terms:
S_24 = (24/2)(a_1 + a_24)
S_24 = 12(7 + (-16))
S_24 = 12(7 - 16)
S_24 = 12(-9)
S_24 = -108
Therefore, the sum of x is 3, and the sum of the first 24 terms of the series is -108.