The sum of first four terms of a linear sequence is 26 and that of the next four terms is 74.find the value of the first term and the common difference

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2(2a + 3D) = 26

2a+3d = 13 **

4(2a + 7d) = 74 + 26 = 100
2a + 7d = 25 ***

Subtract ** from *** to find a and d

Nice one

To find the value of the first term and the common difference in a linear sequence, we can use the formula for the sum of an arithmetic sequence.

The sum of the first four terms of a linear sequence is given as 26. Therefore, we can write the equation:

(1) a + (a + d) + (a + 2d) + (a + 3d) = 26

where "a" represents the first term and "d" represents the common difference.

Similarly, the sum of the next four terms is given as 74. We can write the equation:

(2) (a + 4d) + (a + 5d) + (a + 6d) + (a + 7d) = 74

Now, we have two equations with two variables. We can solve these equations simultaneously to find the values of "a" and "d".

Expanding and simplifying equation (1):
4a + 6d = 26

Expanding and simplifying equation (2):
4a + 22d = 74

Now we can solve these equations.

Multiplying equation (1) by 11:
44a + 66d = 286

Subtracting equation (2) from equation (1):
(44a + 66d) - (4a + 22d) = 286 - 74
40a + 44d = 212

Dividing this equation by 4:
10a + 11d = 53

Now we have a system of two linear equations:
10a + 11d = 53 ...(3)
4a + 6d = 26 ...(4)

Solving equations (3) and (4) simultaneously, we can find the values of "a" and "d".