The sum of the first ten terms of an a.p is 255. Find the sum of the next 20 term of the progression given that the sum of the first twenty term of the progression is 1010
10/2 (2a+9d) = 255
20/2 (2a+19d) = 1010
solve for a and d, and then you want x such that
255 + x = 30/2 (2a+29d)
To find the sum of the next 20 terms of an arithmetic progression (AP) given that the sum of the first 10 terms is 255, we first need to find the common difference (d) and the first term (a₁) of the AP.
The sum of an arithmetic progression can be calculated using the formula: Sn = (n/2)(2a₁ + (n-1)d), where Sn represents the sum of n terms, a₁ is the first term, and d is the common difference.
Given that the sum of the first 10 terms (S₁₀) is 255, we can rewrite the formula as:
255 = (10/2)(2a₁ + (10-1)d)
255 = 5(2a₁ + 9d)
Simplifying further, we have:
51 = 2a₁ + 9d -- (Equation 1)
Now, we need to find the value of a₁ and d using additional information. It is given that the sum of the first 20 terms (S₂₀) is 1010. We can again use the sum formula:
1010 = (20/2)(2a₁ + (20-1)d)
1010 = 10(2a₁ + 19d)
101 = 2a₁ + 19d -- (Equation 2)
We have now formed a system of equations with Equation 1 and Equation 2. We can solve this system of equations to find the values of a₁ and d.
To do this, we'll multiply Equation 1 by 19 and Equation 2 by 9 and subtract them:
19(51) = 19(2a₁ + 9d)
969 = 38a₁ + 171d
9(101) = 9(2a₁ + 19d)
909 = 18a₁ + 171d
Subtracting the equations:
969 - 909 = (38a₁ + 171d) - (18a₁ + 171d)
60 = 20a₁
a₁ = 3
Now, substituting the value of a₁ back into Equation 1:
51 = 2(3) + 9d
51 = 6 + 9d
45 = 9d
d = 5
Therefore, the first term (a₁) is 3, and the common difference (d) is 5.
To find the sum of the next 20 terms, we can substitute the value of a₁ and d in the sum formula:
Sn = (n/2)(2a₁ + (n-1)d)
S₂₀ = (20/2)(2(3) + (20-1)(5))
S₂₀ = 10(6 + 19(5))
S₂₀ = 10(6 + 95)
S₂₀ = 10(101)
S₂₀ = 1010
Thus, the sum of the next 20 terms of the arithmetic progression is 1010.
To find the sum of the next 20 terms of the arithmetic progression (a.p), we need to know the common difference (d) or the first term (a₁) of the progression.
Let's use the sum formula for an arithmetic progression to find the common difference.
The sum of the first ten terms of an a.p can be given by the formula:
S₁₀ = (n/2) * (2a₁ + (n-1)d),
where S₁₀ is the sum of the first ten terms, n is the number of terms, a₁ is the first term, and d is the common difference.
We are given that S₁₀ = 255 and n = 10. Substituting these values into the formula, we get:
255 = (10/2) * (2a₁ + (10-1)d)
255 = 5 * (2a₁ + 9d)
51 = 2a₁ + 9d
Similarly, we can use the sum formula for the first 20 terms of an a.p to find the first term and the common difference.
S₂₀ = (n/2) * (2a₁ + (n-1)d),
where S₂₀ is the sum of the first twenty terms.
We are given that S₂₀ = 1010 and n = 20. Substituting these values into the formula, we get:
1010 = (20/2) * (2a₁ + (20-1)d),
1010 = 10 * (2a₁ + 19d),
101 = 2a₁ + 19d.
Now, we have a system of equations:
51 = 2a₁ + 9d,
101 = 2a₁ + 19d.
By solving this system of equations, we can find the values of a₁ and d:
Multiply the first equation by 2:
102 = 4a₁ + 18d.
Subtract this equation from the second equation:
(2a₁ + 19d) - (4a₁ + 18d) = 101 - 102,
-2a₁ + d = - 1,
2a₁ - d = 1.
Solving these equations simultaneously, we get:
4a₁ + 18d = 102,
2a₁ - d = 1.
Multiplying the second equation by 18 and adding it to the first equation:
4a₁ + 18d + 36a₁ - 18d = 102 + 18,
40a₁ = 120,
a₁ = 3.
Substituting the value of a₁ into the first equation:
2(3) - d = 1,
6 - d = 1,
d = 5.
So, the first term (a₁) is 3, and the common difference (d) is 5.
To find the sum of the next 20 terms, we can use the sum formula again:
S₄₀ = (n/2) * (2a₁ + (n-1)d),
where S₄₀ is the sum of the next twenty terms and n = 20.
Substituting the values of a₁ = 3, d = 5, and n = 20 into the formula, we get:
S₄₀ = (20/2) * (2(3) + (20-1)(5)),
S₄₀ = 10 * (6 + 19 * 5),
S₄₀ = 10 * (6 + 95),
S₄₀ = 10 * 101,
S₄₀ = 1010.
Therefore, the sum of the next twenty terms of the arithmetic progression is 1010.