The first four terms of an arithmetic sequence are
2, x-y, 2x+y+7 and x-3y where x and y are constants.
Find the x and y and hence find the sum of the first 30 terms.
To find the values of x and y, we can use the fact that the terms in an arithmetic sequence have a common difference.
The common difference can be found by subtracting any two consecutive terms. Let's subtract the second term from the first term:
(x - y) - 2 = x - y - 2
Now, let's subtract the third term from the second term:
(2x + y + 7) - (x - y) = x + 2y + 7
Finally, let's subtract the fourth term from the third term:
(x - 3y) - (2x + y + 7) = -x - 4y - 7
Since these differences should all be equal, we can set up a system of equations:
x - y - 2 = x + 2y + 7
x + 2y + 7 = -x - 4y - 7
Simplifying these equations, we get:
-y - 2 = 2y + 7
x + 6y = -14
Solving the first equation for y:
3y = -9
y = -3
Substituting this value into the second equation:
x + 6(-3) = -14
x - 18 = -14
x = -14 + 18
x = 4
Therefore, x = 4 and y = -3.
To find the sum of the first 30 terms, we can use the formula for the sum of an arithmetic sequence:
Sn = (n/2)(2a + (n-1)d)
In this case, the first term (a) is 2, the common difference (d) is x - y, and n is 30.
Sn = (30/2)(2(2) + (30-1)(4-(-3)))
Simplifying,
Sn = 15(4 + 29(7))
Sn = 15(4 + 203)
Sn = 15(207)
Sn = 3105
Therefore, the sum of the first 30 terms is 3105.
To find the values of x and y, we can use the given terms in the arithmetic sequence.
The formula for the nth term of an arithmetic sequence is given by:
an = a1 + (n-1)d
Where:
an is the nth term of the sequence
a1 is the first term of the sequence
n is the position of the term in the sequence
d is the common difference between consecutive terms
Using this formula, we can start by finding the common difference between consecutive terms.
Given the first four terms:
a1 = 2
a2 = x - y
a3 = 2x + y + 7
a4 = x - 3y
The common difference between consecutive terms can be found by subtracting any two consecutive terms. Let's subtract a2 from a1:
a2 - a1 = (x - y) - 2 = x - y - 2
Since this represents the common difference, we can set it equal to the difference between a3 and a2 as well:
a3 - a2 = (2x + y + 7) - (x - y) = 2x + y + 7 - x + y = x + 2y + 7
Setting the two expressions equal to each other:
x - y - 2 = x + 2y + 7
Rearranging and simplifying:
y = -4
Substituting this value of y back into one of the equations, let's use a2:
x - (-4) - 2 = x + 8 - 2 = x + 6
Setting x + 6 equal to 2:
x + 6 = 2
Subtracting 6 from both sides:
x = -4
Hence, the values of x and y are x = -4 and y = -4.
To find the sum of the first 30 terms, we can use the formula for the sum of an arithmetic sequence:
Sn = (n/2)(2a1 + (n-1)d)
Where:
Sn is the sum of the first n terms
a1 is the first term of the sequence
n is the number of terms in the sequence
d is the common difference between consecutive terms
Substituting the values we have:
a1 = 2
n = 30
d = x - y - 2 = -6
Sn = (30/2)(2(2) + (30-1)(-6))
= 15(4 + 174)
= 15(178)
= 2670
Hence, the sum of the first 30 terms is 2670.
(2x+y+7) - (x-y) = (x-y)-2
(x-3y)-(2x+y+7) = (x-y)-2
x=2, y=-3
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