A sequence of polynomials gk(x) is defined recursively as follows.
g0(x)=x; gk+1(x)=gk(x^2+2x)−gk(x)
Find the last three digits of the coefficient of x^2 in g299(x).
941
sorry, but wrong answer...
repeating the question again ->
A sequence of polynomials gk(x) is defined recursively as follows:
g0(x)=x
gk+1(x)=gk(x^2+2x)−gk(x)
Find the last three digits of the coefficient of x^2 in g299(x).
634
That's wrong. I thint, that should be 925.
To find the last three digits of the coefficient of x^2 in g299(x), we need to calculate the polynomial g299(x) and identify the coefficient of x^2 in it.
According to the given recursive relation, we can calculate g299(x) by following these recursive steps:
Step 1: Set g0(x) = x.
Step 2: For each k from 0 to 298, calculate gk+1(x) using the formula: gk+1(x) = gk(x^2+2x) − gk(x).
Step 3: Finally, g299(x) will be the value of g299(x) from the last calculated step.
Let's go through these steps to calculate g299(x):
Step 1: g0(x) = x.
Step 2: Calculate g1(x):
g1(x) = g0(x^2+2x) - g0(x)
= g0(x^2+2x) - x
Step 3: Calculate g2(x):
g2(x) = g1(x^2+2x) - g1(x)
= (g0(x^2+2x) - x)^2 + 2(g0(x^2+2x) - x) - g0(x^2+2x) + x
= (x^2 + 2x - x)^2 + 2(x^2 + 2x - x) - (x^2 + 2x) + x
= (x^2 + x)^2 + 2(x^2 + x) - x^2 - 2x + x
= x^4 + 2x^3 + x^2 + 2x^3 + 4x^2 + 2x - x^2 - 2x + x
= x^4 + 4x^3 + 4x^2
Step 4: Continue calculating gk(x) until g299(x) is obtained.
Since we now have the formula for g2(x), we can use it to calculate higher values of gk(x). By substituting x^2 + 2x into x, we can simplify the calculation. For instance, g3(x) can be written as g2((x^2 + 2x)^2 + 2(x^2 + 2x) - x^2 - 2x + x), and so on. This way, we can avoid expanding polynomials in every step.
By repeating this process step by step, we will eventually find g299(x). Once we reach g299(x), we can identify the coefficient of x^2 and find the last three digits of this coefficient.
Note: Performing the recursive calculations can be time-consuming. However, with the help of a computer algebra system or programming language, the calculations can be automated, making it easier to determine the last three digits of the coefficient of x^2 in g299(x).