Question

Find the number of negative terms in the sequence :
${x_n} = \dfrac{{{}_{}^{n + 4}{P_4}}}{{{P_{n + 2}}}} - \dfrac{{143}}{{4{P_n}}}$

Answer 1

In the given question the term ${P_n}$ and ${P_{n + 2}}$ stands for ${}^{n + 2}{P_{n + 2}}$ and ${}^n{P_n}$ respectively. Therefore , we know that for any permutation expression ${}^n{P_n} = P!$. Now solve it using a simple factorial rule.

Complete step by step answer:
We can write the given equation also as : -
${x_n} = \dfrac{{{}_{}^{n + 4}{P_4}}}{{{}^{n + 2}{P_{n + 2}}}} - \dfrac{{143}}{{4{}^n{P_n}}}$
Now using the formula of permutation
${}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$
Substituting the values we get,
${x_n} = \dfrac{{\dfrac{{(n + 4)!}}{{(n - 4 - 4)!}}}}{{(n + 2)!}} - \dfrac{{143}}{{4n!}}$
$\Rightarrow {x_n} = \dfrac{{\dfrac{{(n + 4)!}}{{n!}}}}{{(n + 2)!}} - \dfrac{{143}}{{4n!}}$
Further solving the factorial in the numerator we get,
${x_n} = \dfrac{{\dfrac{{(n + 4)(n + 3)(n + 2)(n + 1)n!}}{{n!}}}}{{(n + 2)!}} - \dfrac{{143}}{{4.n!}}$
${x_n} = \dfrac{{(n + 4)(n + 3)(n + 2)(n + 1)}}{{(n + 2)!}} - \dfrac{{143}}{{4.n!}}$

Further solving the factorial in the denominator we get,
${x_n} = \dfrac{{(n + 4)(n + 3)(n + 2)(n + 1)}}{{(n + 2)(n + 1)n!}} - \dfrac{{143}}{{4.n!}}$
$\Rightarrow {x_n} = \dfrac{{(n + 4)(n + 3)}}{{n!}} - \dfrac{{143}}{{4.n!}}$
On taking L . C . M , we get
${x_n} = \dfrac{{(4{n^2} + 28n - 95)}}{{4.n!}}$
Now , since we have to find negative terms in ${x_n}$ , so ${x_n}$ will be less zero .
Therefore , ${x_n} = \dfrac{{(4{n^2} + 28n - 95)}}{{4.n!}} < 0$
${x_n} = (4{n^2} + 28n - 95) < 0$ , for solving the value of $n$ we factorize the expression.
The given could be satisfied for the values $n = 1,2$.

Therefore , on putting values of $n$ in equation , we get
$\therefore {x_1} = - \dfrac{{63}}{4}$ and ${x_2} = - \dfrac{{23}}{8}$ .

So, there are two negative values for the given sequence.

Note: If $n$ and $r$ are positive integer such that $1 \leqslant r \leqslant n$ , then the number of all permutation of $n$ distinct things , taken $r$ at a time is denoted by symbol $P(n,r)$ or ${}^n{P_r}$. It should be noted that in permutations , the order of arrangement is taken into account when the order is changed , a different permutation is obtained. In permutations , if there are two jobs such that one of them can be completed in $n$ ways , and when it has been completed in any one of these $n$ ways , second job can be completed in $r$, then the two jobs succession can be completed in $n \times r$ ways.