Find the 19th term of the A.P. $\frac{5}{6}$, $\frac{8}{6}$, $\frac{11}{6}$.................

Analyze the question: We have an arithmetic progression (A.P.) with first term a = 5/6, and we need the 19th term.

Concept: The nth term of an A.P. is

Tₙ = a + (n − 1)d

where d is the common difference.

Solution:

1. First term: a = 5/6
2. Common difference: d = 8/6 − 5/6 = 3/6 = 1/2
3. Number of terms: n = 19

T₁₉ = 5/6 + (19 − 1)(1/2) T₁₉ = 5/6 + 18 × (1/2) T₁₉ = 5/6 + 9 T₁₉ = 9 5/6

Check: Using sixths, the terms are 5/6, 8/6, 11/6, ..., and the numerator goes up by 3 each time. The 19th numerator is 5 + 18 × 3 = 59, so T₁₉ = 59/6 = 9 5/6. ✓

Tip: In A.P. questions, always find d first by subtracting consecutive terms. Remember that you add (n − 1) differences, not n. A common mistake is using 19d instead of 18d, which would give 5/6 + 9.5 = 10 1/3, a value that is not among the options. If your answer isn't among the options, recheck the (n − 1).