Binomial Theorem

Eg 1. Expand $\left(x^2+\frac{3}{x}\right)^4$, $x\ne0$

Sol. By the binomial theorem,

Here, $a=x^2$, $b=\frac{3}{x}$ and $n=4$.

Substituting the values,

Eg 2. Compute $(98)^5$.

Sol. We write $98=100+(-2)$, so $(98)^5=[100+(-2)]^5$.

By the binomial theorem,

Here, $a=100$, $b=-2$ and $n=5$.

Substituting the values,

Eg 3. Which is larger $(1.01)^{1000000}$ or 10,000?

Sol. We write $1.01=1+0.01$, so $(1.01)^{1000000}=(1+0.01)^{1000000}$.

By the binomial theorem,

Here, $a=1$, $b=0.01$ and $n=1000000$.

Since $b>0$, every term of the expansion is positive.

Substituting the values,

Since all the remaining terms are positive,

Hence, $(1.01)^{1000000}$ is the larger number.

Eg 4. Using binomial theorem, prove that $6^n-5n$ always leaves remainder 1 when divided by 25.

Sol. To show that $6^n-5n$ leaves remainder 1 when divided by 25, we have to show that

where $k$ is a whole number.

We write $6=1+5$, so

By the binomial theorem,

Here, $a=1$ and $b=5$.

Substituting the values,

Taking $5^2$ common,

where $k$ is a whole number.

For $n=1$,

Since $0\leq1<25$, the remainder is 1.

∴ $6^n-5n$ always leaves remainder 1 when divided by 25.