Find the exact value (either decimal or fractional 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.
We find a from a system:
a +a+99d =7099.5
a+100d+a+199d=20799
We subtract from the 2nd the 1st:
200d=13699.5
d=68.4975
2a+99x68.4975=7099.5
a=159.12375 (0.12375=99/800)
Ummm.
Sn = n(a1 + an)/2
S100 = 100(a1 + a100)/2
= 50(a+a+99d)
= 100a + 4950
To find the first term of an arithmetic sequence, we need to use the formulas for the sum of the first n terms of an arithmetic sequence.
The sum of the first n terms of an arithmetic sequence is given by the formula:
S(n) = (n/2)(a + l)
where S(n) is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.
In this case, we are given the sum of the first 100 terms (S(100)), which is 7099.5, and the sum of the next 100 terms (S(200)), which is 20799.
Using the formula for S(n), we can set up two equations:
S(100) = (100/2)(a + l) -- Equation 1
S(200) = (200/2)(a + l) -- Equation 2
Substituting the given values, we have:
7099.5 = (100/2)(a + l) -- Equation 1
20799 = (200/2)(a + l) -- Equation 2
Simplifying the equations, we get:
3549.75 = 50(a + l) -- Equation 1 simplified
20799 = 100(a + l) -- Equation 2 simplified
Now we can solve these two equations simultaneously to find a, the first term.
Dividing Equation 2 by 100, we get:
207.99 = a + l -- Equation 3
Substituting Equation 3 into Equation 1, we have:
3549.75 = 50(207.99) -- Equation 1 rewritten with Equation 3
Simplifying Equation 1, we get:
3549.75 = 10399.5
Dividing both sides by 10399.5, we find:
a = 0.34
So the exact value of the first term of the arithmetic sequence is 0.34.