Answer:
[tex]ax^5+ by^5=241[/tex]
Step-by-step explanation:
Given:
- [tex]ax + by = 1[/tex]
- [tex]ax^2+ by^2 = 11[/tex]
- [tex]ax^3+ by^3 = 25[/tex]
- [tex]ax^4+ by^4 = 83[/tex]
We can re-write the left sides of the given equations as follows:
[tex]ax^2+ by^2=(ax+by)(x+y)-xy(a+b)[/tex]
[tex]ax^3+ by^3=(ax^2+by^2)(x+y)-xy(ax+by)[/tex]
[tex]ax^4+ by^4=(ax^3+by^3)(x+y)-xy(ax^2+by^2)[/tex]
Therefore, following this pattern:
[tex]ax^5+ by^5=(ax^4+by^4)(x+y)-xy(ax^3+by^3)[/tex]
Use the given values and the expanded expressions to create 2 equations to help find the values of (x+y) and xy:
Equation 1
[tex]ax^3+ by^3=(ax^2+by^2)(x+y)-xy(ax+by)[/tex]
[tex]\implies 25=11(x+y)-xy(1)[/tex]
[tex]\implies 25=11(x+y)-xy[/tex]
Equation 2
[tex]ax^4+ by^4=(ax^3+by^3)(x+y)-xy(ax^2+by^2)[/tex]
[tex]\implies 83=25(x+y)-xy(11)[/tex]
[tex]\implies 83=25(x+y)-11xy[/tex]
Multiply Equation 1 by 11:
[tex]\implies 275=121(x+y)-11xy[/tex]
Then subtract Equation 2 from this to eliminate 11xy and find the value of (x+y):
[tex]\implies 192=96(x+y)[/tex]
[tex]\implies (x+y)=2[/tex]
Multiply Equation 1 by 25:
[tex]\implies 625=275(x+y)-25xy[/tex]
Multiply Equation 2 by 11:
[tex]\implies 913=275(x+y)-121xy[/tex]
Subtract the 2nd from the 1st to eliminate 275(x+y) and find the value of xy:
[tex]\implies 288=-96xy[/tex]
[tex]\implies xy=-3[/tex]
Therefore, we now have:
- [tex]ax^4+ by^4 = 83[/tex]
- [tex]ax^3+ by^3 = 25[/tex]
- [tex](x+y)=2[/tex]
- [tex]xy=-3[/tex]
Substitute these into the equation for ax⁵ + by⁵ and solve:
[tex]\implies ax^5+ by^5=(ax^4+by^4)(x+y)-xy(ax^3+by^3)[/tex]
[tex]\implies ax^5+ by^5=(83)(2)-(-3)(25)[/tex]
[tex]\implies ax^5+ by^5=166+75[/tex]
[tex]\implies ax^5+ by^5=241[/tex]
