Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Definite Integration

Olympiad workbook · IOQM · Definite Integration

What is the greatest integer not exceeding the sum ∑_ n=1 ^ 1599 1 n

What is the greatest integer not exceeding the sum ∑n=115991n\sum_{n=1}^{1599} \frac{1}{\sqrt{n}}

Correct answer

78

Step-by-step solution

∫11600dxx<∑n=115991n<1+∫11599dxx∣2x∣11600<∑n=115991n<1+∣2x∣1159978<∑n=115991n<21599−178<∑n=115991n<78.97…..⇒[∑n=115991n]=78\begin{aligned} & \int_1^{1600} \frac{d x}{\sqrt{x}} < \sum_{n=1}^{1599} \frac{1}{\sqrt{n}} < 1+\int_1^{1599} \frac{d x}{\sqrt{x}} \\ & |2 \sqrt{x}|_1^{1600} < \sum_{n=1}^{1599} \frac{1}{\sqrt{n}} < 1+|2 \sqrt{x}|_1^{1599} \\ & 78 < \sum_{n=1}^{1599} \frac{1}{\sqrt{n}} < 2 \sqrt{1599}-1 \\ & 78 < \sum_{n=1}^{1599} \frac{1}{\sqrt{n}} < 78.97 \ldots . . \\ & \Rightarrow\left[\sum_{n=1}^{1599} \frac{1}{\sqrt{n}}\right]=78\end{aligned}$$
Practice Definite Integration on Quantrex Academy →

More from Definite Integration

What is the greatest integer not exceeding the sum ∑_ n=1 ^ 1599 1 n Olympiad workbook Full Definite Integration list All Olympiad workbook PYQs