MHB Find x_1+x_3+x_5+--------+x_999

  • Thread starter Thread starter Albert1
  • Start date Start date
AI Thread Summary
The problem involves a sequence of equations where each pair of consecutive variables sums to 1, and the total sum of all variables from x_1 to x_999 equals 999. Given that there are 999 variables, the equations imply that each variable can be expressed in terms of its neighbors. The task is to find the sum of the odd-indexed variables, specifically x_1, x_3, x_5, and so on, up to x_999. The solution requires recognizing the pattern in the sums and applying algebraic manipulation to derive the final result.
Albert1
Messages
1,221
Reaction score
0
$x_1+x_2=x_2+x_3=x_3+x_4=x_4+x_5=--------=x_{998}+x_{999}=1$

$x_1+x_2+x_3+x_4+x_5+--------+x_{999}=999$

find :

$x_1+x_3+x_5+x_7+---+x_{999}=?$
 
Mathematics news on Phys.org
Albert said:
$x_1+x_2=x_2+x_3=x_3+x_4=x_4+x_5=--------=x_{998}+x_{999}=1$

$x_1+x_2+x_3+x_4+x_5+--------+x_{999}=999$

find :

$x_1+x_3+x_5+x_7+---+x_{999}=?$

Given
$x_1+x_2=x_2+x_3=x_3+x_4=x_4+x_5=\cdots=x_{998}+x_{999}=1\cdots(1)$
$x_1+x_2+x_3+x_4+x_5+\cdots+x_{999}=999\cdots(2)$
we have $x_1+x_2=x_2+x_3$ => $x_1= x_3$ ( from (1)
proceeding this way
$x_1 = x_3 = x_ 5 = \cdots = x_{999}\cdots(3)$
and similarly
$x_2 = x_4 = x_ 6 =\cdots = x_{998}\cdots(4)$
further from given relations
$x_1+ x_2 = 1\cdots(5) $ (from(1)
and $500x_1 + 499x_2 = 999\cdots(6) $ from (2), (3) and (4)
multiply (5) by 499 and subtract from (6) giving
$x_1 = 500$
so $x_1+x_3+x_5+x_7+---+x_{999}=500^2 = 250000$
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top