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Explain Polynomial Expansions and Formulating Sequences

Please explain these expansions

Solution
20.
(2x + 3)^4
=(2x)^4 + 4*((2x)^3)*(3) + 6*((2x)^2)*(3^2) + 4*(2x)*(3^3) + 3^4
=16x^4 + 96x^3 + 216x^2 + 216x + 81

21.
(x^2 + 5)^4
=x^4 + 4*(x^3)*(5) + 6*(x^2)*(5^2) + 4*x*(5^3) + 5^4
=x^4 + 20x^3 + 150x^2 + 500x + 625

24.
1,1/4, 1/9, 1/16, 1/25,.....

=> an = 1/(n^2)

25.
-2/3, 3/4, -4/5, 5/6, -6/7,.....

=> an = ((-1)^n)*(1+n)/(2+n)

Solution Preview

20. (2x + 3)^4 :

we know that (x+y) ^2 = x^2 + 2*x*y + y^2
so (x+y)^4 =[ (x+y) ^2] * [ (x+y) ^2 ] = (x^2 + 2*x*y + y^2 ) * (x^2 + 2*x*y + y^2 ) = x^4 + 4*x^3y+ 6*x^2*y^2 + 4xy^3 + y^4,

substitute 2x for x and 3 for y into the above equation and then you can get the result as =16x^4 + 96x^3 + 216x^2 + 216x + 81

21.
similarly, ...

Solution Summary

Polynomial expansions are explained and sequences are formulated.

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