Q.Find the value of $(2+\sqrt{5})^5 + (2-\sqrt{5})^5$.

The Binomial Theorem for a positive integral index gives a single formula for expanding $(a+b)^n$ for any natural number $n$, without multiplying the bracket out by hand $n$ times: $(a+b)^n = {}^nC_0\,a^nb^0 + {}^nC_1\,a^{n-1}b^1 + {}^nC_2\,a^{n-2}b^2 + \cdots + {}^nC_n\,a^0b^n,$ where the coefficients $^nC_0,{}^nC_1,\ldots,{}^nC_n$ are exactly the numbers that appear in Pascal's triangle. The theorem itself is proved using the Principle of Mathematical Induction, with Pascal's rule ($^kC_r+{}^kC_{r-1}={}^{k+1}C_r$) doing the work of the inductive step. A few structural facts follow immediately from the formula and are worth remembering as a checklist: the expansion always has exactly $n+1$ terms; the first term is $a^n$ and the last is $b^n$; in every term the exponents of $a$ and $b$ add up to $n$; moving along the expansion, the exponent of $a$ steadily falls by one while that of $b$ steadily rises by one; and the coefficients read the same from either end (they are symmetric), which mirrors the left-right symmetry of Pascal's triangle. For the subtraction case, $(a-b)^n$ has the same coefficients but with alternating signs, starting positive. This theorem underlies a large family of exa …