Decide whether each expression can be the \(n\)th term of an AP. Justify.
(i) \(2n-3\)
(ii) \(3n^2+5\)
(iii) \(1+n+n^2\)
(i) Yes; (ii) No; (iii) No.
(i) Expression: \(2n - 3\)
Step 1: Write first few terms by putting values of \(n\).
For \(n=1\): \(2(1) - 3 = -1\)
For \(n=2\): \(2(2) - 3 = 1\)
For \(n=3\): \(2(3) - 3 = 3\)
So the terms are: \(-1, 1, 3, 5, ...\)
Step 2: Find the difference between consecutive terms.
\(1 - (-1) = 2,\; 3 - 1 = 2,\; 5 - 3 = 2\)
The difference is always 2 (same each time). So it is an AP.
(ii) Expression: \(3n^2 + 5\)
Step 1: Write first few terms.
For \(n=1\): \(3(1)^2 + 5 = 8\)
For \(n=2\): \(3(2)^2 + 5 = 17\)
For \(n=3\): \(3(3)^2 + 5 = 32\)
For \(n=4\): \(3(4)^2 + 5 = 53\)
So the terms are: \(8, 17, 32, 53, ...\)
Step 2: Find the differences.
\(17 - 8 = 9,\; 32 - 17 = 15,\; 53 - 32 = 21\)
The difference keeps changing (9, 15, 21, ...). Since it is not the same, it is not an AP.
(iii) Expression: \(1 + n + n^2\)
Step 1: Write first few terms.
For \(n=1\): \(1 + 1 + 1^2 = 3\)
For \(n=2\): \(1 + 2 + 2^2 = 7\)
For \(n=3\): \(1 + 3 + 3^2 = 13\)
For \(n=4\): \(1 + 4 + 4^2 = 21\)
So the terms are: \(3, 7, 13, 21, ...\)
Step 2: Find the differences.
\(7 - 3 = 4,\; 13 - 7 = 6,\; 21 - 13 = 8\)
The difference is not the same (4, 6, 8, ...). So it is not an AP.