Excel in math, science, and engineering

New user? Sign up

Existing user? Sign in

ax+by=3 ; ax^2+by^2=7 ; ax^3+by^3=16 ; ax^4+bx^4=42 ; Then find ax^5+bx^5 given a,b,x,y are real nos.

Note by Avinash Panneerselvam 4 years, 3 months ago

Sort by:

Solution : Note that , for any \( n >0 , (ax^n+by^n)(x+y)-xy(ax^{n-1}+by^{n-1}) = ax^{n+1}+by^{n+1}\)

So now For \( n=2 \) we get \( 7(x+y)-3xy=16 \) and for \( n=3\) we get \( 16(x+y)-7xy=42 \) . Solving gives , \( x+y=-14 , xy=-38 \)

now again using the identity ,

\( ax^5+by^5 = (42)(-14)-(16)(-38) = \boxed{20} \) – Shivang Jindal · 4 years, 3 months ago

Log in to reply

a/(ax - 1) + b/(bx - 1) = a + b . Find x . – Himanshu Singh · 2 years ago

Is it \(ax^4 + by^4 = 42, ax^5 + by^5\)? I think you typed wrongly. – Zi Song Yeoh · 4 years, 3 months ago

@Zi Song Yeoh – If it is, then the answer should be \(20\). – Zi Song Yeoh · 4 years, 3 months ago

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

TopNewestSolution : Note that , for any \( n >0 , (ax^n+by^n)(x+y)-xy(ax^{n-1}+by^{n-1}) = ax^{n+1}+by^{n+1}\)

So now For \( n=2 \) we get \( 7(x+y)-3xy=16 \) and for \( n=3\) we get \( 16(x+y)-7xy=42 \) .

Solving gives , \( x+y=-14 , xy=-38 \)

now again using the identity ,

\( ax^5+by^5 = (42)(-14)-(16)(-38) = \boxed{20} \) – Shivang Jindal · 4 years, 3 months ago

Log in to reply

a/(ax - 1) + b/(bx - 1) = a + b . Find x . – Himanshu Singh · 2 years ago

Log in to reply

Is it \(ax^4 + by^4 = 42, ax^5 + by^5\)? I think you typed wrongly. – Zi Song Yeoh · 4 years, 3 months ago

Log in to reply

– Zi Song Yeoh · 4 years, 3 months ago

If it is, then the answer should be \(20\).Log in to reply