In an AP, (a=5), (d=7), and (a_n=103). What is the value of (n)?

Correct answer: (15)

The direct answer is option C, 15. In an arithmetic progression, consecutive terms change by the same common difference d. The nth-term formula is a_n=a+(n-1)d, where a is the first term. Here a=5, d=7, and a_n=103. Substitute these values: 103=5+(n-1)7. Subtract 5 from both sides: 98=7(n-1). Divide by 7: 14=n-1. Add 1: n=15. Therefore, 103 is the fifteenth term. Option A, 13, would give the term 5+12(7)=89, not 103. Option B, 14, would give 5+13(7)=96, so it is one term too early. Option C, 15, gives 5+14(7)=5+98=103, exactly as required. Option D, 16, would give 5+15(7)=110, which is too large. The common mistake is to divide 98 by 7 and stop at 14; that number is n-1, not n. Always add 1 after finding n-1. A useful check is to list the pattern: the first term is 5, and reaching 103 requires fourteen jumps of 7, so the term number is fourteen jumps plus the first term, namely 15.