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
The following figures show three curves generated using an iterative algorithm. The total length of the curve generated after 'Iteration n' is:
Note: The figures shown are representative.
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)
- In the sequence 6, 9, 14, $x$, 30, 41, a possible value of $x$ is
The sum of the first $n$ terms in the sequence 8, 88, 888, 8888, ... is______.
Find the missing group of letters in the following series:
BC, FGH, LMNO, ____________
