List of top Mathematics Questions on Principle of Mathematical Induction

Sum of the series \(\frac {1}{1-3.1^2+1^4}+\frac {2}{1-3.2^2+2^4}+\frac {3}{1-3.3^2+3^4}\) upto \(10\) terms is _____ .
If n \(\in\) [10,100] and n \(\in\) N, then how many such n are possible where 3n - 3 is divisible by 7?
The value of the sum of 10 term of the series
\(S_{10}=\frac{1}{2^2-1}+\frac{1}{4^2-1}+\frac{1}{6^2-1}+...\)
Let \(P(n)=3^{2n+1}+2^{n+2}\) where n∈N. Then
If \(f(z)=\frac{1}{z^2-3z+2}\)is expanded in the region |z|<1, then
The given series \(\frac{x}{1.3}+\frac{x^2}{2.4}+\frac{x^3}{3.5}+......,(x\gt0)\)is convergent in the interval
The infinite series \(\displaystyle\sum_{n=1}^{∞} (1+\frac{1}{n})^{-n^2}\) is:
The infinite series \(\displaystyle\sum_{n=1}^{∞} \frac{3^n}{4^{n+2}}\) is:
Match List I with List II
LIST I LIST II
A.Series \(\displaystyle\sum_{n=1}^{∞} \frac{1}{n^\frac{3}{2}}\) isI.Monotone and 
convergent both
B.Series \(\displaystyle\sum_{n=1}^{∞} \frac{3^n}{n^2}\) isII.\(e^{-2}\)
C.\(\lim\limits_{n \to \infty} (\frac{n+1}{n+2})^{2n+1}\)III.Divergent to ∞
D.sequence \(x_n=1+\frac{1}{2!}+\frac{1}{3!}+…\frac{1}{n!}\) for n∈NIV.Convergent
Choose the correct answer from the options given below:
The value of \(\lim\limits_{n \to \infty} \frac{1}{n} [1+2^{\frac{1}{2}}+3^{\frac{1}{3}}+...n^{\frac{1}{n}}]\) is
If \(\frac {1}{2⋅3^{10}}+\frac {1}{2^2⋅3^9}+⋯+\frac {1}{2^{10}⋅3} = \frac {K}{2^{10}⋅3^{10}}.\) Then the remainder when \(K\) is divided by \(6\) is:
The greatest integer less than or equal to the sum of first \(100\) terms of the sequence \(\frac 13, \frac 59, \frac {19}{27}, \frac {65}{81},…...\) is equal to ______.