If ^n+5P_n+1=- 11(n-1) 2 ^n+3P_n find n. - Vidyadip

Question

If $^\text{n+5}\text{P}_\text{n+1}=-\frac{11(\text{n-1})}{2}\ ^\text{n+3}\text{P}_\text{n}$ find n.

✓

Answer

We have,
$^\text{n+5}\text{P}_\text{n+1}=-\frac{11(\text{n-1})}{2}\ ^\text{n+3}\text{P}_\text{n}$
$\Rightarrow \frac{(\text{n+5)}!}{[\text{n+5}-(\text{n+1})]!}=\frac{11(\text{n-1})}{2}\times\frac{(\text{n}+3)!}{[\text{n}+3-\text{n}]!}$
$\Rightarrow \frac{(\text{n+5)}!}{[\text{n+5}-\text{n-1}]!}=\frac{11(\text{n-1})}{2}\times\frac{(\text{n}+3)}{3!}$
$\Rightarrow \frac{(\text{n+5)}!}{4!}=\frac{11(\text{n-1})}{2}\times\frac{(\text{n}+3)!}{3!}$
$\Rightarrow \frac{(\text{n+5)}(\text{n+4)}(\text{n+3)}!}{4!}=\frac{11(\text{n-1})}{2}\times\frac{(\text{n}+3)!}{3!}$
$\Rightarrow \frac{(\text{n+5)}(\text{n+4)}!}{4\times3!}=\frac{11(\text{n-1})}{2\times3}$
$\Rightarrow (\text{n}+5)(\text{n}+4)=\frac{11(\text{n}-1)\times4}{2}$
$\Rightarrow (\text{n}+5)(\text{n}+4)=22(\text{n}-1)$
$\Rightarrow \text{n}^2+4\text{n}+5\text{n}+20=22\text{n}-22$
$\Rightarrow \text{n}^2+9\text{n}-22\text{n}+20+22=0$
$\Rightarrow \text{n}^2-13\text{n}+42=0$
$\Rightarrow \text{n}^2-6\text{n}-7\text{n}+42=0$
$\Rightarrow \text{n}(\text{n}-6)-7(\text{n}-6)=0$
$\Rightarrow \text{n}=6 \ \text{or},\ \text{n}=7$
Hence, $\text{n} =6\ \text{or},\ 7$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Permutations→Permutations — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the coefficients of a⁴in the product $(1+2\text{a})^{4}(2-\text{a})^{5}$ using binomial theorem.

If A, B, C are three sets such that $\text{A}\subset\text{B},$ then prove that $\text{C} - \text{B}\subset\text{C} - \text{A}.$

Prove that $\frac{1}{2}\tan\Big(\frac{\text{x}}{2}\Big)+\frac{1}{4}\tan\Big(\frac{\text{x}}{4}\Big)+...+\frac{1}{2^{\text{n}}}\tan\Big(\frac{\text{x}}{2^{\text{n}}}\Big)=\frac{1}{2^{\text{n}}}\cot\Big(\frac{\text{x}}{2^{\text{n}}}\Big)-\cot\text{x}$ for all $\text{n}\in\text{N}$ and $0<\text{x}<\frac{\pi}{2}$

Find the mean deviation from the mean for the following data:

classes	95-105	105-115	115-125	125-135	135-145	145-155
Frequencies	9	13	16	26	30	12

If the line y = mx + 1 is tangent to the parabola y² = 4x, then find the value of m.

If ${^\text{n}}\text{C}_{\text{4}},{^\text{n}}\text{C}_{\text{5}}$ and ${^\text{n}}\text{C}_{\text{6}}$ are in A.P. then find n.

Evaluate:
$\lim\limits_{\text{x} \rightarrow 0}\frac{2\sin\text{x}-\sin2\text{x}}{\text{x}^{3}}$

Find the equation of the straight line which has y-intercept equal to $\frac{4}{3}$ and is perpendicular to 3x - 4y + 11 = 0.

Evaluate the following limit:
$\lim\limits_{\text{n}\rightarrow\infty}{\Big(\frac{1}{\text{n}^2}+\frac{2}{\text{n}^2}+\frac{3}{\text{n}^2}+\ \cdots+\frac{\text{n}-1}{\text{n}^2}}{}\Big)$

Calculate the mean deviation about the median of the following observation:
3011, 2780, 3020, 2354, 3541, 4150, 5000