The sum of 16th term is 240,the sum of the next four terms is 220.Find the first term and the common difference.
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To find the solution, we need to use the formulas for the sum of an arithmetic sequence.
The formula to find the sum of an arithmetic sequence is:
Sn = (n/2)(2a + (n-1)d)
Where:
Sn is the sum of the first n terms
a is the first term
d is the common difference
n is the number of terms
From the given information, we have the following equations:
1) S16 = 240 (the sum of the first 16 terms is 240)
2) S20 = 220 (the sum of the next four terms is 220)
We can start by substituting the known values into the sum formula.
For equation 1:
240 = (16/2)(2a + (16-1)d)
240 = 8(2a + 15d)
For equation 2:
220 = (20/2)(2a + (20-1)d)
220 = 10(2a + 19d)
Now we have a system of two equations with two unknowns (a and d).
Simplifying equation 1:
240 = 8(2a + 15d)
240 = 16a + 120d
Simplifying equation 2:
220 = 10(2a + 19d)
220 = 20a + 190d
Now we can solve this system of equations simultaneously using either substitution or elimination.
Let's solve using elimination:
Multiply equation 1 by 10 and equation 2 by 8 to eliminate the variable 'a'.
2400 = 80a + 1200d
1760 = 160a + 1520d
Now subtract equation 1 from equation 2:
440 = 80a + 320d
Divide both sides by 40:
11 = 2a + 8d -(equation 3)
Now we have two equations:
440 = 80a + 320d -(equation 4)
11 = 2a + 8d -(equation 3)
Multiply equation 3 by 40:
440 = 80a + 320d -(equation 4)
440 = 80a + 320d -(equation 3)
Subtract equation 3 from equation 4:
0 = 0
Since we obtained an identity (0 = 0), this means the equations are dependent, and there are infinitely many solutions.
This implies that it is not possible to determine the first term (a) and the common difference (d) based on the given information.