Binomial Theorem — Practice

Question 1

Find the term independent of $x$ in $\left(\dfrac{3x^{2}}{2}-\dfrac{1}{3x}\right)^{9}$.

Accepted Answer

$T_{r+1}=\binom{9}{r}\left(	frac32ight)^{9-r}(-	frac13)^{r}x^{18-2r-r}$.

$18-3r=0$ gives $r=6$.

$\binom{9}{6}\left(	frac32ight)^{3}\left(-	frac13ight)^{6}=84	imes	frac{27}{8}	imes	frac{1}{729}=	frac{7}{18}$.

Question 2

Find the coefficient of $x^{4}$ in $(1+x)^{10}+(1+x)^{11}+\cdots+(1+x)^{20}$.

Accepted Answer

The sum equals $\dfrac{(1+x)^{21}-(1+x)^{10}}{x}$.

So the coefficient of $x^{4}$ is the coefficient of $x^{5}$ in the numerator: $\binom{21}{5}-\binom{10}{5}$.

$=20349-252=20097$.

Question 3

In $(1+x)^{20}$, which term is greatest when $x=\dfrac{4}{5}$?

Accepted Answer

$\dfrac{(n+1)x}{1+x}=\dfrac{21	imes\frac45}{\frac95}=\dfrac{84}{9}=9.33$.

So the terms increase while $r\le9$, and the greatest is at $r=9$, the tenth term.

The greatest coefficient, by contrast, is $\binom{20}{10}$ at the eleventh term.

Question 4

How many rational terms are there in the expansion of $\left(\sqrt[4]{5}+\sqrt[6]{2}\right)^{24}$?

Accepted Answer

$T_{r+1}$ carries $5^{(24-r)/4}\,2^{r/6}$.

Need $4\mid(24-r)$ and $6\mid r$, so $4\mid r$ and $6\mid r$, hence $12\mid r$.

$r=0,12,24$: three rational terms.

Question 5

If the coefficients of the $5$th, $6$th and $7$th terms of $(1+x)^{n}$ are in arithmetic progression, find $n$.

Accepted Answer

$2\binom{n}{5}=\binom{n}{4}+\binom{n}{6}$.

Dividing by $\binom{n}{5}$: $2=\dfrac{5}{n-4}+\dfrac{n-5}{6}$.

Multiplying out: $12(n-4)=30+(n-5)(n-4)$, so $n^{2}-21n+98=0$.

$n=7$ or $n=14$.

Question 6

Evaluate $\displaystyle\sum_{r=0}^{n}(-1)^{r}\binom{n}{r}$ and $\displaystyle\sum_{r=0}^{n}\frac{(-1)^{r}}{r+1}\binom{n}{r}$.

Accepted Answer

Putting $x=-1$ in $(1+x)^{n}$ gives $0$ for $n\ge1$.

Integrating $(1-x)^{n}$ from $0$ to $1$: $\displaystyle\int_0^1(1-x)^n\,dx=\frac{1}{n+1}$.

Expanding the integrand term by term gives the stated sum, so it equals $\dfrac{1}{n+1}$.

Question 7

Prove that $\displaystyle\sum_{r=0}^{n}r\binom{n}{r}^{2}=n\binom{2n-1}{n-1}$.

Accepted Answer

$r\binom{n}{r}=n\binom{n-1}{r-1}$, so the sum is $n\sum_r\binom{n-1}{r-1}\binom{n}{r}$.

Writing $\binom{n}{r}=\binom{n}{n-r}$, the sum is $\sum_r\binom{n-1}{r-1}\binom{n}{n-r}$, which by Vandermonde is the coefficient of $x^{n-1}$ in $(1+x)^{2n-1}$.

Hence the total is $n\binom{2n-1}{n-1}$.

Question 8

Find the first three terms of the expansion of $(4-3x)^{1/2}$ and state where it is valid.

Accepted Answer

$(4-3x)^{1/2}=2\left(1-	frac{3x}{4}ight)^{1/2}$, valid for $\left|	frac{3x}{4}ight|<1$, that is $|x|<	frac43$.

$2\left(1+	frac12u+	frac{\frac12(-\frac12)}{2}u^{2}ight)$ with $u=-	frac{3x}{4}$.

$=2-	frac{3x}{4}-	frac{9x^{2}}{64}$.

Question 9

Find the coefficient of $x^{3}$ in $\dfrac{1}{(1-2x)(1-3x)}$, valid for small $x$.

Accepted Answer

$\dfrac{1}{(1-2x)(1-3x)}=\dfrac{3}{1-3x}-\dfrac{2}{1-2x}$.

Coefficient of $x^{3}$: $3\cdot3^{3}-2\cdot2^{3}=81-16=65$.

Valid for $|x|<	frac13$, the stricter of the two conditions.

Question 10

How many terms are there in $(x+y+z+w)^{8}$, and what is the coefficient of $x^{2}y^{2}z^{2}w^{2}$?

Accepted Answer

Number of terms: $\binom{8+3}{3}=\binom{11}{3}=165$.

Coefficient: $\dfrac{8!}{2!\,2!\,2!\,2!}=\dfrac{40320}{16}=2520$.

Question 11

Find the remainder when $7^{103}$ is divided by $25$.

Accepted Answer

$7^{2}=49=50-1$, so $7^{102}=(50-1)^{51}$.

Modulo $25$, every term with a factor of $50$ vanishes, leaving $(-1)^{51}=-1$.

So $7^{102}\equiv-1$ and $7^{103}\equiv-7\equiv18$ modulo $25$.

Question 12

Show that $2^{4n}-15n-1$ is divisible by $225$ for every positive integer $n$.

Accepted Answer

$2^{4n}=16^{n}=(1+15)^{n}=1+15n+\binom{n}{2}15^{2}+\cdots$

Subtracting $15n+1$ removes exactly the first two terms.

Every remaining term carries $15^{2}=225$, so the whole expression is divisible by $225$.

Practice — Binomial Theorem

Sign in to save attempts to your dashboard and track your accuracy over time

Find a verified tutor

Join a live course

Browse coaching centres

About

Categories

Support

Documentation

Get Study Tips & Updates

Download Our App

Download for iOS

Download for Android

Popular Cities