Understanding Complex Number Magnitude
We are given a complex number $z$ defined as a product of terms: $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$ where $i = \sqrt{-1}$. We are also given that the square of its magnitude is $|z|^2 = 44200$. The objective is to find the value of $n$.
Modulus Property for Products
A key property of complex number magnitudes is that the magnitude of a product is the product of the magnitudes.
Therefore, $|z| = |1 + i| \cdot |1 + 2i| \cdot |1 + 3i| \dots |1 + ni|$.
The magnitude of a complex number $a + bi$ is calculated as $|a + bi| = \sqrt{a^2 + b^2}$.
Applying this, we get:
$|z| = \sqrt{1^2 + 1^2} \cdot \sqrt{1^2 + 2^2} \cdot \sqrt{1^2 + 3^2} \dots \sqrt{1^2 + n^2}$
This simplifies to:
$|z| = \sqrt{(1+1^2)(1+2^2)(1+3^2) \dots (1+n^2)}$
Calculating |z|^2
Squaring both sides gives the magnitude squared:
$|z|^2 = (1+1^2)(1+2^2)(1+3^2) \dots (1+n^2)$
We are given $|z|^2 = 44200$. So, we need to find $n$ such that:
$ \prod_{k=1}^{n} (1+k^2) = 44200 $
Step-by-Step Calculation
Let's compute the product for successive values of $n$:
- For $n=1$: $|z|^2 = (1+1^2) = 2$
- For $n=2$: $|z|^2 = (1+1^2)(1+2^2) = 2 \times (1+4) = 2 \times 5 = 10$
- For $n=3$: $|z|^2 = (1+1^2)(1+2^2)(1+3^2) = 10 \times (1+9) = 10 \times 10 = 100$
- For $n=4$: $|z|^2 = (1+1^2)(1+2^2)(1+3^2)(1+4^2) = 100 \times (1+16) = 100 \times 17 = 1700$
- For $n=5$: $|z|^2 = (1+1^2)(1+2^2)(1+3^2)(1+4^2)(1+5^2) = 1700 \times (1+25) = 1700 \times 26 = 44200$
The calculated value for $n=5$ matches the given $|z|^2 = 44200$.
Conclusion
The value of $n$ that satisfies the condition $|z|^2 = 44200$ is $n=5$.
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 $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. 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 ___________.
- If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of $\left(\frac{1}{x^3} - x^4\right)^n$, $x \neq 0$, is zero, then the value of $n$ is _________.
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 $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. 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 ____________.
