If the 6th term of an A.P. is equal to four times its first term and the sum of fist six terms is 75, find the first term and the common difference.

For an A.P.: T₆ = a+5d.

Given T₆ = 4a: a+5d = 4a 5d = 3a …(1)

Also: S₆ = 6/2[2a+5d] = 75 3(2a+5d)=75 2a+5d=25 …(2)

From (1): a = 5d/3.

Substitute in (2): 10d/3 + 5d = 25 25d/3 = 25 d = 3.

Then: a = 5.

Answer: First term = 5; common difference = 3.

Use the sixth-term condition to relate a and d, then use the sum of the first six terms.