The third term of an A.P. is 6 and the fifth term is 12. Find the sum of its first twelve terms

Step 1: Write A.P. term formula
nth term: aₙ = a + (n – 1)d
Where a = first term, d = common difference

Step 2: Set up equations
Third term: a + 2d = 6 … (1)
Fifth term: a + 4d = 12 … (2)

Step 3: Find common difference
Subtract equation (1) from (2):
(a + 4d) – (a + 2d) = 12 – 6
2d = 6
d = 3

Step 4: Find first term
Substitute d = 3 into equation (1):
a + 2(3) = 6
a + 6 = 6
a = 0

Step 5: Calculate sum of 12 terms
Sₙ = (n/2)[2a + (n – 1)d]
S₁₂ = (12/2)[2(0) + (12 – 1)(3)]
= 6[0 + 11 × 3]
= 6 × 33
= 198