IGNOU MTE 4 2026 - Show that .b) Let . Show that

Question

Show that $1 + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{n}} > 2\sqrt{n + 1} - 2, \forall n \in \mathbb{N}$ .
b) Let $a, b > 0, a + b = 1, n > 1$ . Show that $\left(a + \frac{1}{a}\right)^n + \left(b + \frac{1}{b}\right)^n \ge \frac{5^n}{2^{n-1}}$

09 Jan 2026

Answer :

Word Count : 115

For (a), consider the function (f(x)=\frac{1}{\sqrt{x}}), which is positive and decreasing on ([1,\infty)). For each integer (k\ge1), [ \int_k^{k+1}\frac{dx}{\sqrt{x}}<\frac{1}{\sqrt{k}}. ] Hence, [ \sum_{k=1}^n\frac{1}{\sqrt{k}}> \int_1^{n+1}\frac{dx}{\sqrt{x}}. ] Evaluating the integral, [ \int_1^{n+1}x^{-1/2}dx=\left[2\sqrt{x}\right]_1^{n+1}=2\sqrt{n+1}-2. ] Therefore, [ 1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}}>2\sqrt{n+1}-2, ] which proves the required inequality for all (n\in\mathbb{N}). For (b),

__________ ____ _______ _____ _________ ___ ______ __________ ______ ___ _______ ______.
___ _____ ________ ________ __________ _________ ______ ____ ______ ______ ______ ______.
__________ _______ ___ _____ _______ ____ _______.
______ ____ ____ ____ _______ _____ _____ _______ _______.
___ ________ ___ __________ _________.
_______ _____ ____ ______ ________ _____ __________ ___ ____ _____ ________ __________.
____ __________ _________ ____ ______ ____ ________ ______.
_____ _____ _____ ________ ________.

IGNOU ACADEMY

Your IGNOU journey, made easier.