In the AP (75,98,121,\ldots), how many terms are less than (2500)?

Correct answer: 106

Here, the first term is \(a=75\) and the common difference is \(d=98-75=23\). The \(n\)th term is \(a_n=75+23(n-1)\). Using \(75+23(n-1)<2500\), we get \(n-1<\frac{2425}{23}\approx105.43\). Hence, the greatest integer value of \(n\) is \(106\). Therefore, 106 terms are less than 2500. For \(n=107\), the term is \(2513\), which is not less than 2500. Exam tip: retain the strict inequality \((<)\) when the question says “less than.”