General Term Analysis
The series is given by \( \frac{1}{1+1^{4} \times 4} + \frac{2}{1+2^{4} \times 4} + \frac{3}{1+3^{4} \times 4} + \dots \). The general term, \( T_k \), is:
\( T_k = \frac{k}{1+4k^4} \)
Factor the denominator using the identity \( 1+4x^4 = (1+2x^2+2x)(1+2x^2-2x) \). Substituting \( x=k \):
\( 1+4k^4 = (2k^2 + 2k + 1)(2k^2 - 2k + 1) \)
We express \( T_k \) using the difference of two fractions. Note that \( (2k^2 + 2k + 1) - (2k^2 - 2k + 1) = 4k \).
\( T_k = \frac{1}{4} \times \frac{4k}{(2k^2 - 2k + 1)(2k^2 + 2k + 1)} \) \( T_k = \frac{1}{4} \left( \frac{1}{2k^2 - 2k + 1} - \frac{1}{2k^2 + 2k + 1} \right) \)
Telescoping Series Summation
Define \( f(k) = 2k^2 - 2k + 1 \). Then \( f(k+1) = 2(k+1)^2 - 2(k+1) + 1 = 2(k^2+2k+1) - 2k - 2 + 1 = 2k^2 + 2k + 1 \).
The general term simplifies to a telescoping form: \( T_k = \frac{1}{4} \left( \frac{1}{f(k)} - \frac{1}{f(k+1)} \right) \)
The sum of the first 10 terms, \( S_{10} \), is: \( S_{10} = \sum_{k=1}^{10} T_k = \frac{1}{4} \sum_{k=1}^{10} \left( \frac{1}{f(k)} - \frac{1}{f(k+1)} \right) \) This sum telescopes to: \( S_{10} = \frac{1}{4} \left( \frac{1}{f(1)} - \frac{1}{f(11)} \right) \)
Calculating the Sum \( \frac{m}{n} \)
First, compute \( f(1) \) and \( f(11) \): \( f(1) = 2(1)^2 - 2(1) + 1 = 1 \) \( f(11) = 2(11)^2 - 2(11) + 1 = 2(121) - 22 + 1 = 242 - 21 = 221 \)
Now, calculate \( S_{10} \): \( S_{10} = \frac{1}{4} \left( \frac{1}{1} - \frac{1}{221} \right) = \frac{1}{4} \left( \frac{221 - 1}{221} \right) = \frac{1}{4} \times \frac{220}{221} = \frac{55}{221} \)
So, \( \frac{m}{n} = \frac{55}{221} \).
Finding \( m+n \)
We have \( m = 55 \) and \( n = 221 \). Check for the greatest common divisor (GCD): \( m = 55 = 5 \times 11 \) \( n = 221 = 13 \times 17 \)
The prime factors of \( m \) and \( n \) are distinct, so \( \operatorname{gcd}(55, 221) = 1 \).
The value required is \( m+n \): \( m+n = 55 + 221 = 276 \)
Similar Questions
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
- Let $a_1 = 1$ and for $n \ge 1, a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}$. Then $\left| \sum_{n=1}^\infty \left( a_n - \frac{2}{n^2} \right) \right|$ is equal to ____________.
If $ \left(\frac{1}{^{15}C_0} + \frac{1}{^{15}C_1}\right)$ $\left(\frac{1}{^{15}C_1} + \frac{1}{^{15}C_2}\right)$ $\dots$ $ \left(\frac{1}{^{15}C_{12}} + \frac{1}{^{15}C_{13}}\right)$ = $\frac{\alpha^{13}}{^{14}C_0 ^{14}C_1 \dots ^{14}C_{12}} $, then $30\alpha$ is equal to __________.
- Three persons enter in a lift at the ground floor. The lift will go upto 10$^{\text{th}}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to _________.
- The number of 4-letter words, with or without meaning, which can be formed using the letters PQR PQRSTUVP, is ___________.
- Let $|A| = 6$, where A is a $3 \times 3$ matrix. If $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot 3^n$, $m, n \in \mathbf{N}$, then $m + n$ is equal to ___________.
Important Questions from Algebra
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
