the sum of the first four terms is 26 and that of the next four terms is 74. find the values of (a) the first terms (b) the common difference.
To find the values of the first term and the common difference, we can use the formula for the sum of an arithmetic series.
The formula for the sum of an arithmetic series is:
Sn = (n/2)(2a + (n-1)d),
where:
- Sn represents the sum of the series
- a represents the first term of the series
- d represents the common difference of the series
- n represents the number of terms in the series
Let's use this formula to solve the problem.
Given:
1) The sum of the first four terms is 26, so S4 = 26
2) The sum of the next four terms is 74, so S8 = 74
We can calculate the values of the first term (a) and the common difference (d) using these two equations:
Equation 1: S4 = (4/2)(2a + (4-1)d) = 26
Equation 2: S8 = (8/2)(2a + (8-1)d) = 74
Simplifying the equations:
Equation 1: 2(2a + 3d) = 26
Equation 2: 4(2a + 7d) = 74
Expanding and simplifying further:
Equation 1: 4a + 6d = 13
Equation 2: 8a + 28d = 37
We have two equations with two variables. Now we can solve this system of equations.
Multiplying Equation 1 by 4 and subtracting it from Equation 2:
(8a + 28d) - 4(4a + 6d) = 37 - 4(13)
8a + 28d - 16a - 24d = 37 - 52
-8a + 4d = -15
Simplifying further:
-8a + 4d = -15
Now, we can solve for one variable in terms of the other:
-8a = -4d - 15
Divide both sides by -8:
a = (4d + 15)/8
Now, substitute this value of a in either equation 1 or equation 2 and solve for d.
Substitute a into Equation 1:
4(4d + 15)/8 + 6d = 13
4(4d + 15) + 48d = 104
16d + 60 + 48d = 104
64d + 60 = 104
64d = 44
d = 44/64
d = 11/16
Now that we have the value of d, substitute it back into Equation 1 to find a:
4a + 6(11/16) = 13
4a + 66/16 = 13
4a + 66 = 208
4a = 208 - 66
4a = 142
a = 142/4
a = 71/2
Therefore, the values of the first term (a) and the common difference (d) are:
a = 71/2
d = 11/16
Assuming we have an AS
sum(4) = 26 ---> 2(2a + 3d) = 26
2a + 3d = 13
sum(8) = 26+74 =100 --> 4(2a + 7d) = 100
2a + 7d = 50
subtract : 4d = 37
d = 37/4
in 2a + 3d = 13
2a + 111/4 = 13
a = -59/8