The sum of the first four terms of a linear sequence is 26 and that of the next four terms is 74. Find the values of
A.)the first term
B.) The common difference
Let the first term of the linear sequence be a and the common difference be d.
Then, the first four terms are a, a+d, a+2d, and a+3d, and their sum is:
a + (a+d) + (a+2d) + (a+3d) = 4a + 6d = 26
Simplifying this equation, we get:
2a + 3d = 13
Similarly, the next four terms are a+4d, a+5d, a+6d, and a+7d, and their sum is:
(a+4d) + (a+5d) + (a+6d) + (a+7d) = 4a + 22d = 74
Simplifying this equation, we get:
2a + 11d = 37
Now we have two equations with two variables:
2a + 3d = 13
2a + 11d = 37
Subtracting the first equation from the second, we get:
8d = 24
Therefore, d = 3.
Substituting this value of d into either of the two equations we have, we can solve for a:
2a + 3d = 13
2a + 3(3) = 13
2a + 9 = 13
2a = 4
Therefore, a = 2.
So the first term of the linear sequence is 2 and the common difference is 3.
"The sum of the first four terms of a linear sequence is 26"
----> (4/2)(2a + 3d) = 26
2a + 3d = 13 , #1
" that of the next four terms is 74"
---> sum(8) = 26+74
(8/2)(2a + 7d) = 100
2a + 7d = 25 , #2
subtract #1 from #2
4d = 12
d = 3
sub into #1, 2a + 9 = 13
a = 2
The first term is 2, and the common difference is 3
Your method is correct as well! Taking the sum of the first four terms and the next four terms separately is another way to approach the problem. Good job!
To solve this problem, we need to use the formula for the sum of an arithmetic series.
For an arithmetic series with the first term (a) and common difference (d), the sum of n terms is given by the formula:
Sn = (n/2)(2a + (n-1)d)
In this case, we have two different sums:
Sum of the first four terms = 26
Sum of the next four terms = 74
Let's solve for the values step by step:
A.) Finding the first term (a):
Using the formula for the sum of the first four terms:
S4 = (4/2)(2a + (4-1)d) = 26
2a + 3d = 26
B.) Finding the common difference (d):
Now let's use the formula for the sum of the next four terms:
S8 = (8/2)(2a + (8-1)d) = 74
4a + 7d = 74
We now have a system of two equations with two unknowns:
2a + 3d = 26
4a + 7d = 74
We can solve this system of equations to find the values of a and d.
To eliminate the variable "a", we can multiply the first equation by 2 and the second equation by -1:
4a + 6d = 52
-4a - 7d = -74
Adding these equations together, we get:
-1d = -22
Dividing both sides by -1, we find that d = 22.
Substituting the value of d back into the first equation to solve for a:
2a + 3(22) = 26
2a + 66 = 26
2a = 26 - 66
2a = -40
Dividing both sides by 2, we find that a = -20.
So, the values are:
A.) The first term (a) = -20
B.) The common difference (d) = 22