# Telescopic Series

$~~~\displaystyle ~~\sum_{k=4}^{n} \{~~f(k-2) - f(k)~~\}$
$=~~\displaystyle \sum_{k=4}^{n}f(k-2) - \sum_{k=4}^{n}f(k)$
$=~~\displaystyle \sum_{k=2}^{n-2}f(k) - \sum_{k=4}^{n}f(k)$
$=~~\displaystyle f(2) + f(3) +\sum_{k=4}^{n-2}f(k) - \sum_{k=4}^{n-2}f(k) - f(n-1) - f(n)$
$=~~\displaystyle f(2) + f(3) + ............... 0 ............................. - f(n-1) - f(n)$

$\displaystyle NOTE:-Change~of~boundareys$ $\displaystyle \sum_{k=m}^{n}f(k) = \sum_{k=m+p}^{n+p}f(k-p)…p~is~any~intiger.$
$\displaystyle \sum_{k=m}^{n}f(k) = \sum_{k=m}^{r}f(k) + \sum_{k=r+1}^{n}f(k)->splitting~the~range~into~~m~to~r~~and~~~(r+1)~to~ n.$

We change the first summation from 4 through n to 2 through n-2 so that the expression inside becomes X^k for both summation. Then separate out summations between k=4 and k=n-2 which cancels out to 0. We are left with remaining first TWO terms from the first summation and last TWO from the second, since there is a difference of 2 (p=2) in the start of the two summations. For other p values this should be adjusted suitably Note by Niranjan Khanderia
6 years, 7 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

Um... Consider using latex. All this "^" stuff makes it really confusing.

Cheers

- 6 years, 5 months ago

Thanks. I have now used \displaystyle \color\red{Latex}

- 6 years, 4 months ago