Find the exact value (either decimal or fraction form) of the first term of an arithmetic sequence if the sum of the first 100 terms is 7099.5 and the sum of the next 100 terms is 20799.
To find the exact value of the first term of an arithmetic sequence, we need to know the common difference. However, the problem does not directly provide the common difference.
Instead, we can use the formulas to find the sum of an arithmetic sequence:
The sum of the first n terms of an arithmetic sequence can be calculated using the formula:
Sn = (n/2)(2a + (n-1)d)
Where:
Sn is the sum of the first n terms,
a is the first term, and
d is the common difference.
Since the sum of the first 100 terms is 7099.5, and the sum of the next 100 terms is 20799, we can set up two equations:
For the first 100 terms:
7099.5 = (100/2)(2a + (100-1)d) ...(Equation 1)
For the next 100 terms:
20799 = (100/2)(2a + (100-1)d) ...(Equation 2)
Since both equations have the same values for the common difference (d), we can subtract Equation 1 from Equation 2 to eliminate the variable d:
20799 - 7099.5 = 100a + (100-1)d - (100a + (100-1)d)
13699.5 = 100d
Simplifying, we find that d = 136.995.
Substituting this value of d in Equation 1, we can solve for a:
7099.5 = (100/2)(2a + (100-1)(136.995))
Simplifying further:
7099.5 = 50(2a + 13699.5)
7099.5 = 100a + 68497.5
100a = -61498
Dividing by 100, we find that a = -614.98.
Therefore, the first term of the arithmetic sequence (either in decimal or fraction form) is -614.98.