Calculating Difference: Sum of Numbers
We need to find the difference between the sum of the first $2n$ natural numbers and the sum of the first $n$ odd natural numbers.
Sum of First $2n$ Natural Numbers
The formula for the sum of the first $k$ natural numbers is $\frac{k(k+1)}{2}$. For the first $2n$ natural numbers, we set $k = 2n$. Sum $= \frac{(2n)(2n+1)}{2} = n(2n+1) = 2n^2 + n$.
Sum of First $n$ Odd Natural Numbers
The sum of the first $n$ odd natural numbers ($1, 3, 5, \dots, 2n-1$) is given by the formula $n^2$.
Finding the Difference
The difference is calculated as:
- (Sum of first $2n$ natural numbers) - (Sum of first $n$ odd natural numbers)
- $ (2n^2 + n) - (n^2) $
- $ 2n^2 + n - n^2 $
- $ n^2 + n $
Therefore, the difference is $n^2 + n$.
Important Questions from Series
- In the sequence 6, 9, 14, $x$, 30, 41, a possible value of $x$ is
Let $a_0 = 0$ and define $a_n = \frac{1}{2}(1 + a_{n-1})$ for all positive integers $n \ge 1$.
The least value of $n$ for which $|1 - a_n| < \frac{1}{2^{10}}$ is __________.
(Answer in integer)
- Calculate the reciprocal of the coefficient of $z^3$ in the Taylor series expansion of the function $f(z) = \sin(z)$ around $z = 0$. (Provide the answer as an integer.)
- Let $a_1 = 1$ and $a_n = a_{n-1} + 4$, $n \ge 2$. Then,
$\lim_{n\to\infty} \left[\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \dots + \frac{1}{a_{n-1}a_n}\right]$
is equal to ________ - Let $S(x) = a_0 + \sum_{n=1}^\infty(a_n \cos (n x) + b_n \sin (n x))$ be the Fourier series of the$2 \pi$ periodic function defined by $f(x) = x^2 + 4 \sin (x) \cos(x)$, $-\pi \le x \le \pi$. Then
$|\sum_{n=0}^\infty a_n - \sum_{n=1}^\infty b_n|$
is equal to ________
