# Thread: Arccot series and Arithmetic progressions?

1. ## Arccot series and Arithmetic progressions?

If $\displaystyle x_1,x_2,x_3,\mbox{...}$ are in arithmetic progression with common difference $\displaystyle d$, show that:

$\displaystyle \mbox{arccot}\left(\frac{1 + x_{1}x_{2}}{d}\right) + \mbox{arccot}\left(\frac{1 + x_{2}x_{3}}{d}\right) + \mbox{arccot}\left(\frac{1 + x_{3}x_{4}}{d}\right) + \mbox{...}$ $\displaystyle +\ \mbox{arccot}\left(\frac{1 + x_{n}x_{n + 1}}{d}\right) = \mbox{arccot}\left(\frac{1 + x_{1}x_{n + 1}}{d}\right)$, where $\displaystyle x_1 > 0$ and $\displaystyle d > 0$

2. ## This statement is false.

Letting $\displaystyle x=\{2,4,6,8,10,12\}$ ,

$\displaystyle \sum_{k=1}^5 \arctan(\frac{2}{1+x_kx_{k+1}}) = .381$

but $\displaystyle \arctan(\frac{2}{1+x_1x_{6}}) = .0798$

3. Originally Posted by fardeen_gen
If $\displaystyle x_1,x_2,x_3,\mbox{...}$ are in arithmetic progression with common difference $\displaystyle d$, show that:

$\displaystyle \mbox{arccot}\left(\frac{1 + x_{1}x_{2}}{d}\right) + \mbox{arccot}\left(\frac{1 + x_{2}x_{3}}{d}\right) + \mbox{arccot}\left(\frac{1 + x_{3}x_{4}}{d}\right) + \mbox{...}$ $\displaystyle +\ \mbox{arccot}\left(\frac{1 + x_{n}x_{n + 1}}{d}\right) = \mbox{arccot}\left(\frac{1 + x_{1}x_{n + 1}}{d}\right)$, where $\displaystyle x_1 > 0$ and $\displaystyle d > 0$
Since they are in A.P with common difference 'd', $\displaystyle x_2 - x_1 = x_3 - x_2 = x_4 - x_3 = \cdots = d$

$\displaystyle \sum_{k=1}^{k=n}\mbox{arccot}\left(\frac{1 + x_{k}x_{k+1}}{d}\right) = $$\displaystyle \sum_{k=1}^{k=n}\mbox{arccot}\left(\frac{1 + x_{k}x_{k+1}}{x_{k+1} - x_k}\right)$$\displaystyle = \sum_{k=1}^{k=n}\mbox{arccot}(x_{k+1}) - \mbox{arccot}(x_{k}) = \mbox{arccot}(x_{n+1}) - \mbox{arccot}(x_1)$$\displaystyle =\mbox{arccot}\left(\frac{1 + x_{1}x_{n+1}}{nd}\right)$

4. ## Slight Correction

Isomorphism:

$\displaystyle \mbox{arccot}(x_{n+1}) - \mbox{arccot}(x_1)$ = $\displaystyle \mbox{arccot}\left(\frac{1 + x_{1}x_{n+1}}{x_{n+1}-x_{1}}\right)$ = $\displaystyle \mbox{arccot}\left(\frac{1 + x_{1}x_{n+1}}{nd}\right)$

This slight correction will make the theorem true.