Using the general term

The general term T(k+1) = nCk a^(n−k) b^k, finding a particular coefficient, the term independent of x, and the greatest coefficient using the ratio of consecutive terms.

The general term

A particular coefficient

Example 1●●

Find the coefficient of $x^5$ in $(2x - 1)^8$.

$T_{k+1} = {}^8C_k\,(2x)^{8-k}(-1)^k = {}^8C_k\,2^{8-k}(-1)^k\,x^{8-k}$

$8 - k = 5$, so $k = 3$.

Coefficient: ${}^8C_3 \times 2^5 \times (-1)^3 = 56 \times 32 \times (-1) = -1792$

The term independent of x

Example 2●●●

Find the term independent of $x$ in $\left(x^2 + \dfrac2x\right)^6$.

$T_{k+1} = {}^6C_k\,(x^2)^{6-k}\left(\dfrac2x\right)^k = {}^6C_k\,2^k\,x^{12 - 2k - k} = {}^6C_k\,2^k\,x^{12 - 3k}$

$12 - 3k = 0$, so $k = 4$.

${}^6C_4 \times 2^4 = 15 \times 16 = 240$

The greatest coefficient

To find the largest coefficient, compare each coefficient $t_k$ with the next one, using the ratio $\dfrac{t_{k+1}}{t_k}$. The coefficients increase while the ratio is greater than $1$.

Example 3●●●

Find the greatest coefficient in the expansion of $(1 + 2x)^7$.

$t_k = {}^7C_k\,2^k$, so $\dfrac{t_{k+1}}{t_k} = \dfrac{{}^7C_{k+1}}{{}^7C_k} \times 2 = \dfrac{2(7 - k)}{k + 1}$.

$\dfrac{2(7 - k)}{k + 1} > 1$ when $14 - 2k > k + 1$, that is $k < \dfrac{13}{3}$, so for $k = 0, 1, 2, 3, 4$.

Greatest: $t_5 = {}^7C_5 \times 2^5 = 21 \times 32 = 672$.

The coefficients rise up to t₅ and then fall.

Flashcards

General term of $(a + b)^n$

$T_{k+1} = {}^nC_k\,a^{n-k}b^k$

Term independent of $x$

Set the total power of $x$ to $0$ and solve for $k$.

$dfrac{{}^nC_{k+1}}{{}^nC_k}$

$\dfrac{n - k}{k + 1}$

Practice

Find the coefficient of $x^5$ in $(2x - 1)^8$.
Find the term independent of $x$ in $\left(x^2 + \dfrac2x\right)^6$.
Find the greatest coefficient in $(1 + 2x)^7$.
In $(a + b)^n$, the general term $T_{k+1}$ is:
For $\left(x^2 + \dfrac2x\right)^6$, which value of $k$ gives the constant term?
Find the coefficient of $x^2$ in $\left(x + \dfrac1x\right)^5$.
Find the coefficient of $x^3$ in $(1 + 2x)^6$.
$\dfrac{{}^nC_{k+1}}{{}^nC_k} = $
How many terms does the expansion of $(1 + 2x)^6$ have?
Find the term independent of $x$ in $\left(x - \dfrac2x\right)^6$.