- AYes, it is the 54th term
- BNo, because n does not come out to be a whole number
- CYes, it is the 55th term
- DNo, because an AP can never contain negative terms
Step-by-Step Solution
a=11,d=−3, so an=11−3(n−1)=14−3n. Setting 14−3n=−150 gives 3n=164,n=3164, which is not a whole number, so -150 is not a term. It slips between a54=−148 and a55=−151. (Negative terms themselves are perfectly allowed.)
Answer: B) No, because n does not come out to be a whole number
Key idea: n-th term formula
**aₙ = a + (n − 1)d**
Where a is the first term, d is the common difference, n is the position.
**Why**: a₁ = a; a₂ = a + d; a₃ = a + 2d; … aₙ = a + (n − 1)d. Each step adds another d.
**Example**: AP 3, 7, 11, 15, …. Find the 20th term.
a = 3, d = 4. a₂₀ = 3 + 19·4 = 79.
**Reverse use**: 'In the AP 5, 11, 17, …, which term is 89?' a + (n−1)d = 89 → 5 + 6(n−1) = 89 → n = 15.