Question

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

Answer 1

"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

Answer 2

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.

Answer 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!