All Questions
20
questions
141
votes
36
answers
308k
views
Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
- 1,585
11
votes
1
answer
13k
views
Sum of series : $1+11+111+...$
Sum of series $1+11+111+\cdots+11\cdots11$ ($n$ digits)
We have:
$1=\frac {10-1}9,$
$11=\frac {10^2-1}9$
.
.
.
$11...11= \frac {10^n-1}9$ (number with $n$ digits)
and summing them we find the ...
- 11.3k
9
votes
8
answers
3k
views
The sum of consecutive integers is $50$. How many integers are there?
I started off by calling the number of numbers in my list "$n$". Since the integers are consecutive, I had $x + (x+1) + (x+2)...$ and so on. And since there were "$n$" numbers in my list, the last ...
7
votes
3
answers
12k
views
Sum of first n natural numbers proof
I know how to prove this by induction but the text I'm following shows another way to prove it and I guess this way is used again in the future. I'm confused by it.
So the expression for first n ...
- 639
4
votes
3
answers
119
views
Find value of $a_{2012}$
A sequence $\left\{a_n\right\}$ is defined as:
$a_1=1$, $a_2=2$ and
$$a_{n+1}=\frac{2}{a_n}+a_{n-1}$$ $\forall$ $n \ge 2$
Find $a_{2012}$
My Try:
we have
$$a_{n+1}-a_{n-1}=\frac{2}{a_n}$$
$$...
- 13.2k
2
votes
2
answers
103
views
Find the value of $\sum_{p=0}^{\infty}\sum_{q=0}^{\infty} \frac{2^{-p-q}}{1+p+q}$
Find the value of
$$S=\sum_{p=0}^{\infty}\sum_{q=0}^{\infty} \frac{2^{-p-q}}{1+p+q}$$
In the second summation i used change of variable $p+q+1=r$ then we get
$$S=\sum_{p=0}^{\infty}\:\sum_{r=p+1}^...
- 13.2k
2
votes
1
answer
2k
views
Arithmetic Series: find the sum of 25 terms given 2 terms and their values
I need help with this question:
"Find S25, given an arithmetic series whose 8th term is 16 and whose 13th term is 81."
What I did was:
1. Found the common difference (d) like this:
81 - 16 = 65
...
1
vote
3
answers
981
views
Sum of all the numbers in the grid.
A square containing numbers
$$
\begin{array}{|c|c|c|}
\hline 1 & 2 & 3 \\ \hline
1 & 2 & 2 \\ \hline
1 & 1 & 1 \\\hline
\end{array}
\qquad \qquad\qquad
\begin{array}{|c|c|c|c|...
1
vote
2
answers
54
views
Proof of a series involving Arithmetic Progression
If $${a}_{1},{a}_{2},{a}_{3},.....{a}_{n-1},{a}_{n} $$ are in A.P., then show that $$ \frac{1}{{a}_{1}{a}_{n}} + \frac{1}{{a}_{2}{a}_{n-1}} +\frac{1}{{a}_{3}{a}_{n-2}} +.....+\frac{1}{{a}_{n}{a}_{1}} =...
- 1,319
1
vote
3
answers
55
views
Step in proof of derivation of $1+2+\cdots+n=\tfrac{n(n+1)}{2}$.
I have been solving a few computer science problems lately and it is important for me to understand how the time complexity of an algorithm is calculated by coming up with the derivation and arriving ...
- 113
0
votes
4
answers
762
views
How does one evaluate $1+2-3-4+5+6-7-8+\cdots+50$?
How does one evaluate the sum $1+2-3-4+5+6-7-8+\cdots+50$?
I know how to find the sum of arithmetic progressions: without the negative signs, one simply has
$$
1+2+\cdots+50=\frac{1}{2}\cdot(1+50)\...
- 109
0
votes
5
answers
334
views
Find the sum to $n$ terms of the following series
Find the sum to $n$ terms of the following series:
$$\dfrac {2}{5}+\dfrac {6}{5^2}+\dfrac {10}{5^3}+\dfrac {14}{5^4}+………$$
My Attempt:
Let
$$S_n=\dfrac {2}{5} + \dfrac {6}{5^2}+\dfrac {10}{5^3}+\...
- 7,426
0
votes
5
answers
1k
views
Pre-calculus- Finding the sum of $1.2+2.3x+3.4x^2...$ where $|x|<1$
Find the sum of $1.2+2.3x+3.4x^2...\infty$ where $|x|<1$
I have got the $\mathbb{n^{th}\quad term \quad= n(n+1)(x^{n-1})}$
Then I tried finding $\sum\limits_{k=1}^\infty n(n+1)(x^{n-1})$ but it's ...
- 6,111
0
votes
2
answers
3k
views
I need help with proofs using mathematical induction: $2+7+12+17+...+(5n-3)=(\frac{n}{2})(5n-1)$
I need help with this problem: $2+7+12+17+...+(5n-3)=(\frac{n}{2})(5n-1)$
- 21
0
votes
4
answers
114
views
Find the sum of the first $50$ terms of the series $a_{n} = -4a_{n-1} + 3$.
I'm not sure where to start this without being given some terms.
Find the sum of the first $50$ terms of the series $$a_{n} = -4a_{n-1} + 3$$ I can see that the common difference is $-4$ and the slope ...
- 172