f\left(n\right)=\left\{\frac{\frac{n}{2}\ \ \ \ if\ n\ is\ even}{3n+1\ \ \ \ \ \ if\ n\ is\ odd}\right\}

The sequence$$1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2,2,2,2,2,1,2,\dots$$consists of $1$'s separated by blocks of $2$'s with $n$ $2$'s i

kicyunya [14]

Consider the lengths of consecutive 1-2 blocks.

block 1 - 1, 2 - length 2

block 2 - 1, 2, 2 - length 3

block 3 - 1, 2, 2, 2 - length 4

block 4 - 1, 2, 2, 2, 2 - length 5

and so on.

Recall the formula for the sum of consecutive positive integers,

$\displaystyle \sum_{i=1}^j i = 1 + 2 + 3 + \cdots + j = \frac{j(j+1)}2 \implies \sum_{i=2}^j = \frac{j(j+1) - 2}2$

Now,

$1234 = \dfrac{j(j+1)-2}2 \implies 2470 = j(j+1) \implies j\approx49.2016$

which means that the 1234th term in the sequence occurs somewhere about 1/5 of the way through the 49th 1-2 block.

In the first 48 blocks, the sequence contains 48 copies of 1 and 1 + 2 + 3 + ... + 47 copies of 2, hence they make up a total of

$\displaystyle \sum_{i=1}^48 1 + \sum_{i=1}^{48} i = 48+\frac{48(48+1)}2 = 1224$

numbers, and their sum is

$\displaystyle \sum_{i=1}^{48} 1 + \sum_{i=1}^{48} 2i = 48 + 48(48+1) = 48\times50 = 2400$

This leaves us with the contribution of the first 10 terms in the 49th block, which consist of one 1 and nine 2s with a sum of $1+9\times2=19$ .

So, the sum of the first 1234 terms in the sequence is 2419.

8 0

2 years ago

Graph this piecewise function:

Pepsi [2]

Consider the function f(x) = 2 x + 1. We recognize the equation y = 2 x + 1 as the Slope-Intercept form of the equation of a line with slope 2 and y-intercept (0,1). Think of a point moving on the graph of f. As the point moves toward the right it rises.
hope this helps

6 0

3 years ago

30 random samples of high school students were asked if they played a sport for their high school. Each sample was 20 students.

Ludmilka [50]

Answer:

a. This option is correct

b. This will be true only if we take the mode as the best representative value

c. This is a correct choice

d. This is not a correct choice

Step-by-step explanation:

Statistics

We are given a dot plot representing 30 random sample proportions of high school students about their sports activities. Based on the data extracted from the plot, we can make some basic conclusions and, less accurately, some predictions.

From the data plot we can see that the following proportions were obtained, along with their absolute frequencies:

0.05 -> 3

0.10 -> 8

0.15 -> 7

0.20 -> 5

0.25 -> 4

0.30 -> 2

Let's select which of the following options are correct

a. A sample proportion of 0.20 means that 4 of 20 high school students responded that they play a sport.

The ratio between the sport playing students to the total of high school students is

$\displaystyle \frac{4}{20}=\frac{1}{5}=0.20$

Thus, this statement is correct.

b. The best prediction for the proportion of all high school students that play a sport is 0.10.

This will be true only if we take the mode as the best representative value for the whole dataset. Personally, I don't like to trust the mode as a good central tendency result, I'd prefer the mean instead.

c. The mean of the 30 sample proportions calculated to the nearest hundredth is 0.16

Computing the mean of the sample proportions

$\displaystyle \bar x=\frac{\sum x_i.f_i}{\sum f_i}$

$\displaystyle \bar x=\frac{0.00\cdot 1+0.05\cdot 3+0.10\cdot 8+0.15\cdot 7+0.20\cdot 5+0.25\cdot 4+0.30\cdot 2}{1+3+8+7+5+4+2}$

$\bar x\approx 0.16$

This is a correct choice

d. The shape of the sample distribution is fairly symmetrical

We can see the distribution if left-skewed because its peak value is not at the mean value. The mode and the mean are fairly different, thus this choice is not correct

5 0

3 years ago

Read 2 more answers

So i need to pick classes for my freshman year next year and i d k if i should take pre algebra or just do algebra. My math grad

ahrayia [7]

Answer:

Step-by-step explanation:

My recommendation is try to find a finals for both pre-algebra or algebra after learning some of the material ahead of time and after taking them, find which one is more comfortable for you

Good luck!

8 0

2 years ago

(rs) (4) = (r/s) (3) =

Dmitriy789 [7]

Answer:

(rs) (4) = (r/s)(3) = (r)(s)(2)

4 0

3 years ago