The sum of the first four terms of a linear sequence (A.P.) is 26 and that of the next four terms is 74. Find the values of:
(1) the first term;
(2) the common difference.
Let the first term of the A.P. be 'a' and the common difference be 'd'.
The sum of the first four terms of the A.P. is 26:
a + (a+d) + (a+2d) + (a+3d) = 26
4a + 6d = 26
2a + 3d = 13
The sum of the next four terms of the A.P. is 74:
(a+4d) + (a+5d) + (a+6d) + (a+7d) = 74
4a + 22d = 74
2a + 11d = 37
Now, we have two equations:
2a + 3d = 13
2a + 11d = 37
Subtracting the first equation from the second:
8d = 24
d = 3
Substitute the value of d back into the first equation:
2a + 3(3) = 13
2a + 9 = 13
2a = 4
a = 2
Therefore, the first term of the A.P. is 2 and the common difference is 3.