The number of positive terms in the sequence xn=1954.nPn− (n+3)P3 (n+1)Pn+1,n∈N is - Sri Chaitanya Infinity Learn Best Online Courses for NCERT Solutions, CBSE, ICSE, JEE, NEET, Olympiad and Class 6 to 12

Solution:

We have

$\begin{array}{l} x_{n} = \frac{195}{{4.}^{n} P_{n}} - \frac{^{(n + 3)} P_{3}}{^{(n + 1)} P_{n + 1}} \\ = \frac{195}{4. n!} - \frac{(n + 3) (n + 2) (n + 1)}{(n + 1)!} \\ = \frac{195}{4. n!} - \frac{(n + 3) (n + 2)}{n!} \\ = \frac{195 - 4 n^{2} - 20 n - 24}{4. n!} \\ x_{n} is positive \end{array}$

$∴ \frac{171 - 4 n^{2} - 20 n}{4. n!} > 0$

$4 n^{2} + 20 n - 171 < 0$ which is true for $n = 1, 2, 3, 4$

$∴$ Given sequence has 4 positive terms

The number of positive terms in the sequence $x_{n} = \frac{195}{{4.}^{n} P_{n}} - \frac{^{(n + 3)} P_{3}}{^{(n + 1)} P_{n + 1}}, n \in N$ is

Solution:

Related content