1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Rzqust [24]
2 years ago
15

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

n the $n^{\rm{th}}$ block. What is the sum of the first $1234$ terms of this sequence
Mathematics
1 answer:
kicyunya [14]2 years ago
8 0

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.

You might be interested in
1. Ruth Lockwood earns $2,917 a month as a supervisor for Acme
soldi70 [24.7K]

Answer:

a. 25%

b. $1560

c. $130

Step-by-step explanation:

a.

If 75% is paid by company, the rest is paid by Ruth.

Out of 100%, 75% is paid by company, so Ruth pays 100 - 75 = 25%

b.

Per year, the total amount paid is 6240, but Ruth pas 25% of this. So,

25% = 0.25

To get the answer, we multiply 0.25 with 6240.

0.25 * 6240 = 1560

So, $1560 is paid by Ruth annually.

c.

Since 1560 is paid annually (in 12 months), each month Ruth has to pay:

1560/12 = $130

Hence, $130 is deducted each month from her paycheck

6 0
3 years ago
Find the greatest common factor.<br> 42²x² + 42²x+ 40²<br> Enter the correct answer.
svlad2 [7]

Answer:

4a^2

Step-by-step explanation:

4a^2x^2+4a^2x+4a^2= \\\\4a^2(x^2+x+1)

Therefore, the largest common factor is 4a^2. Hope this helps!

3 0
3 years ago
Can you guys help me. I will mark you as a brainliest. And give you a really good rating.
Nikitich [7]

Add 5ft 7in and 4ft 8in together

5ft 7in + 4ft 8in = 10ft 3in

6 0
3 years ago
Read 2 more answers
Please answer !!!!!!!
Akimi4 [234]

The measure of all the angles is given below:

angle 1= 30

angle 2=  150

angle 3= 30

angle 4= 150

angle 5= 30

angle 6= 150

angle 7= 30

angle 8= 73.5

<h3>What is angle?</h3>

An angle is formed when two straight lines or rays meet at a common endpoint. The common point of contact is called the vertex of an angle.

As,

angle 1= 30 (Vertically opposite angle)

angle 7= 30 (corresponding angle)

angle 4+30= 180 (linear pair)

 angle 4= 150

angle 4= angle 2= 150 (Vertically opposite angle)

angle 2= angle 6= 150  (corresponding angle)

angle 1= angle 5= 30  (corresponding angle)

angle (2x+3)= angle 4  (corresponding angle)

     2x+3= 150

      x= 73.5

Lean more about angle here:

brainly.com/question/13954458

#SPJ1      

7 0
1 year ago
Find the missing length.
Alex17521 [72]
Can you show the problem
8 0
3 years ago
Other questions:
  • 9 times (30+7=(9 times _)+(9 times 7)<br><img src="https://tex.z-dn.net/?f=9%20%5Ctimes%20%2830%20%2B%207%29%20%3D%20%289%20%5Ct
    13·2 answers
  • 107/13 as a mixed number
    13·2 answers
  • What's the greatest common number between 5,11
    10·1 answer
  • please help me with this i asked last time but it didn’t some the important stuff so i’m just putting a picture ^
    8·1 answer
  • Helppppp
    5·2 answers
  • Without drawing the graph of the function y=1.2x−7, see if the graph passes through points: ( 300; 353)
    14·1 answer
  • Find the length of x
    10·1 answer
  • The chart below lists the original and sale prices of items at a clothing store.
    14·1 answer
  • Which equation represents the relationship between x and y shown in the table?
    11·2 answers
  • Multiply the Polynomials:<br><br> (2x+3) (-x+2y-4)
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!