x+x*1.05+x*1.05^1+x*1.05^2+x*1.05^3+x*1.05^4+x*1.05^5+x*1.05^6+x*1.05^7+x*1.05^8+x*1.05^9+x*1.05^10=1000

Solution for x+x*1.05+x*1.05^1+x*1.05^2+x*1.05^3+x*1.05^4+x*1.05^5+x*1.05^6+x*1.05^7+x*1.05^8+x*1.05^9+x*1.05^10=1000 equation:

x+x*1.05+x*1.05^1+x*1.05^2+x*1.05^3+x*1.05^4+x*1.05^5+x*1.05^6+x*1.05^7+x*1.05^8+x*1.05^9+x*1.05^10=1000

We move all terms to the left:

x+x*1.05+x*1.05^1+x*1.05^2+x*1.05^3+x*1.05^4+x*1.05^5+x*1.05^6+x*1.05^7+x*1.05^8+x*1.05^9+x*1.05^10-(1000)=0

Wy multiply elements

x^2+x^2+x^2+x^2+x^2+x^2+x^2+x^2+x^2+x^2+x+1.05x-1000=0

We add all the numbers together, and all the variables

10x^2+2.05x-1000=0

a = 10; b = 2.05; c = -1000;
Δ = b2-4ac
Δ = 2.052-4·10·(-1000)
Δ = 40004.2025
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(2.05)-\sqrt{40004.2025}}{2*10}=\frac{-2.05-\sqrt{40004.2025}}{20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(2.05)+\sqrt{40004.2025}}{2*10}=\frac{-2.05+\sqrt{40004.2025}}{20}
$