Let {𝑎 𝑛 } 𝑛≥1 be a sequence such that 𝑎 1 =1 and 4𝑎 𝑛+1 = $\sqrt{45 + 16𝑎_𝑛} ,𝑛$ =1, 2, 3, … . Then, which one of the following statements is TRUE?

{𝑎 𝑛 } 𝑛≥1 1 is bounded above by $\frac{17}{8}$

$∑^∞ _ {n=1} a_n$ is convergent

Let {𝑎 𝑛 } 𝑛≥1 be a sequence such that 𝑎 1 =1 and 4𝑎 𝑛+1 = $\sqrt{45 + 16𝑎_𝑛} ,𝑛$ =1, 2, 3, … . Then, which one of the following statements is TRUE?

{𝑎 𝑛 } 𝑛≥1 1 is bounded above by $\frac{17}{8}$

$∑^∞ _ {n=1} a_n$ is convergent

Question:

Let {𝑎_𝑛}_𝑛≥1 be a sequence such that 𝑎₁=1 and 4𝑎_𝑛+1= $\sqrt{45 + 16𝑎_𝑛} ,𝑛$ =1, 2, 3, … . Then, which one of the following statements is TRUE?

IIT JAM MS - 2023
IIT JAM MS

Updated On: Oct 1, 2024

{𝑎_𝑛 }_𝑛≥1 is monotonically increasing and converges to $\frac{17}{8}$
{𝑎_𝑛 }_𝑛≥1 is monotonically increasing and converges to $\frac{9}{4}$
{𝑎_𝑛 }_𝑛≥11 is bounded above by $\frac{17}{8}$
$∑^∞ _ {n=1} a_n$ is convergent

Hide Solution

Verified By Collegedunia

The Correct Option is B

Solution and Explanation

The correct option is (B): {𝑎_𝑛 }_𝑛≥1 is monotonically increasing and converges to

\frac{9}{4}

Was this answer helpful?

Questions Asked in IIT JAM MS exam

Suppose that the weights (in kgs) of six months old babies, monitored at a healthcare facility, have $N(\mu, \sigma^2)$ distribution, where $\mu \in \mathbb{R}$ and $\sigma > 0$ are unknown parameters. Let $X_1, X_2, \ldots, X_9$ be a random sample of the weights of such babies. Let $\overline{X} = \frac{1}{9} \sum_{i=1}^{9} X_i$ , $S = \sqrt{\frac{1}{8} \sum_{i=1}^{9} (X_i - \overline{X})^2}$ and let a 95% confidence interval for $\mu$ based on $t$ -distribution be of the form $(\overline{X} - h(S), \overline{X} + h(S))$ , for an appropriate function $h$ of random variable $S$ . If the observed values of $\overline{X}$ and $S^2$ are 9 and 9.5, respectively, then the width of the confidence interval is equal to __________ (round off to 2 decimal places) (You may use $t_{9,0.025} = 2.262, t_{8,0.025} = 2.306, t_{9,0.05} = 1.833, t_{8,0.05} = 1.86$ ).
- IIT JAM MS - 2024
- Multivariate Distributions
View Solution
Let $X_1, X_2, X_3$ be a random sample from a Poisson distribution with mean $\lambda$ , $\lambda > 0$ . For testing $H_0: \lambda = \frac{1}{8}$ against $H_1: \lambda = 1$ , a test rejects $H_0$ if and only if $X_1 + X_2 + X_3 > 1$ . The power of this test is equal to __________ (round off to 2 decimal places).
- IIT JAM MS - 2024
- Standard Univariate Distributions
View Solution
Let $X_1, X_2, \ldots, X_{10}$ be a random sample from a $U(-\theta, \theta)$ distribution, where $\theta \in (0, \infty)$ . Let $X_{(10)} = \max \{ X_1, X_2, \ldots, X_{10} \}$ and $X_{(1)} = \min \{ X_1, X_2, \ldots, X_{10} \}$ . If the observed values of $X_{(10)}$ and $X_{(1)}$ are 8 and -10, respectively, then the maximum likelihood estimate of $\theta$ is equal to __________ (answer in integer).
- IIT JAM MS - 2024
- Estimation
View Solution
Let $\Theta$ be a random variable having $U(0, 2\pi)$ distribution. Let $X = \cos \Theta$ and $Y = \sin \Theta$ . Let $\rho$ be the correlation coefficient between $X$ and $Y$ . Then $100\rho$ is equal to __________ (answer in integer).
- IIT JAM MS - 2024
- Univariate Distributions
View Solution
Consider a coin for which the probability of obtaining head in a single toss is $\frac{1}{3}$ . Sunita tosses the coin once. If head appears, she receives a random amount of $X$ rupees, where $X$ has the $\text{Exp}\left( \frac{1}{9} \right)$ distribution. If tail appears, she loses a random amount of $Y$ rupees, where $Y$ has the $\text{Exp}\left( \frac{1}{3} \right)$ distribution. Her expected gain (in rupees) is equal to __________ (answer in integer).
- IIT JAM MS - 2024
- Probability
View Solution

View More Questions

Let {𝑎𝑛 }𝑛≥1 be a sequence such that 𝑎1=1 and 4𝑎𝑛+1=45+16𝑎𝑛,𝑛\sqrt{45 + 16𝑎_𝑛} ,𝑛45+16an​​,n=1, 2, 3, … . Then, which one of the following statements is TRUE?

The Correct Option is B

Solution and Explanation

Top Questions on Sequences and Series of real numbers

Questions Asked in IIT JAM MS exam

Let {𝑎_𝑛}_𝑛≥1 be a sequence such that 𝑎₁=1 and 4𝑎_𝑛+1= $\sqrt{45 + 16𝑎_𝑛} ,𝑛$ =1, 2, 3, … . Then, which one of the following statements is TRUE?