Question

a= 1/2 (5-√21) then value of a^3 + a^-3 - 5a^2 - 5a^-2 + a + a^-1

Answer 1

Answer:

(a^3 + a^-3) - 5(a^2 + a^-2) + (a + a^-1)

Assume a^3 as x and a^-3 as y. The 1st segment looks as x^3 + y^3. You know the formula of (x+y)^3 = x^3 + y^3 + 3xy(x+y). So x^3 + y^3 = (x+y)^3 - 3xy(x+y).

Similarly look at 2nd segment which too can be rephrased in the method of (x+y)^2

Doing this, you'll get (a+1/a)^3 - 3(a+1/a) - (a+1/a)^2 + 2 + (a+1/a)

Now let x = a+1/a

Substitute a with the value given in your question and take common denominator and simplify. you'll get the value of x as 5

put this in the equation we got 2 steps back which is x^3 - x^2 - 2x + 2

You'll get the answer as 92