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
kumpel [21]
3 years ago
5

n the sequence $121, 1221, 12221, \dots ,$ the $n$th number consists of $n$ copies of the digit $2$, surrounded by two $1$s. How

many of the first $100$ terms in the sequence are divisible by $3$?
Mathematics
2 answers:
asambeis [7]3 years ago
8 0

Answer:

  33

Step-by-step explanation:

The number is divisible by 3 if the sum of digits is divisible by 3. 1221 is the first number with a sum of digits divisible by 3. Adding 3 more 2s to the sequence will result in another number divisible by 3.

So, every 3rd term from n=2 to n=98 will be divisible by 3, for a total of 33 terms out of the first 100 in the sequence.

balandron [24]3 years ago
7 0

The nth number in the sequence can be expressed as

a_n=10^{n+1}+\displaystyle2\sum_{i=2}^n10^i+21

Extracting the nth term from the sum gives

a_n=10^{n+1}+2\cdot10^n+\displaystyle2\sum_{i=2}^{n-1}10^i+21

and 10^{n+1}+2\cdot10^n=10^n(10+2)=12\cdot10^n. 3 divides both 12 and 21, so 12\cdot10^n and 21 contribute no remainder.

This leaves us with

a_n\equiv10\cdot\underbrace{222\ldots222}_{n-2\text{ copies}}\pmod3

Recall that a decimal integer is divisible by 3 if its digits add to a multiple of 3. The digits in 10\cdot222\ldots222 are n-2 copies of 2 and one 0, so the digital sum is 2(n-2)=2n-4.

  • If n=3k for k=0,1,2,3,\ldots, then the digital sum is 2(3k)-4=6k-4, which is not divisible by 3.
  • If n=3k+1, then the sum is 2(3k+1)-4=6k-2, which is not divisible by 3.
  • If n=3k+2, then the sum is 2(3k+2)-4=6k, which is always divisble by 3.

This means that roughly 1/3 of the first n numbers in this sequence are divisible by 3; among the first 100 terms, they occur for n=2,5,8,\ldots,95,98, of which there are 33.

You might be interested in
Find a polynomial function of least degree having only real coefficients, a leading coefficient of 1, and roots of 2-√6 , 2+ √6
horsena [70]

Answer:

x^{4} -18x^{3}+104x^{2} -172x-100

Step-by-step explanation:

The 3 roots are given out of which 2 are real and 1 is imaginary. For a polynomial of least degree having real coefficients, it must have a complex conjugate root as the 4th root. Therefore, based on 4 roots, the least degree of polynomial will be 4. Finding the polynomial having leading coefficient=1 and solving it based on multiplication of 2 quadratic polynomials, we get:  

\\\\x_{1} = 2-\sqrt{6} \\x_{2} = 2+\sqrt{6} \\x_{3}=7-i \\x_{4}=7+i \\\\P(x)=1(x-x_{1})(x-x_{2} )(x-x_{3} )(x-x_{4} ) \\\\=(x-(2-\sqrt{6}))(  x-(2+\sqrt{6} )) (x-(7-i))( x-(7+i))\\=((x-2)+\sqrt{6})( ( x-2)-\sqrt{6} ) ((x-7)+i)( (x-7)-i)\\=((x-2)^{2} -(\sqrt{6} )^{2} )((x-7)^{2}-(i)^{2})\\=(x^{2} -4x-2)(x^{2} -14x+50)\\=x^{4} -18x^{3}+104x^{2} -172x-100\\

7 0
2 years ago
Whats the y-axis reflection. of 4,-4
Karo-lina-s [1.5K]

Answer:

4,4

Step-by-step explanation:

its ask for the reflection, so -4 will be 4

5 0
2 years ago
If f(x) = 2x + 1 and g(x) = f(x) + 3,<br> what would be the function of g(x)?
Maslowich

f(x) = 2x + 1 and g(x) = f(x) + 3

g(x) = 2x + 1 + 3

<h2><u>g(x) = 2x + 4</u></h2>
3 0
2 years ago
Please help me :((((
Keith_Richards [23]

Answer:

b i think

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
BRAINLIEST Create a wrapping paper design that includes at least four of the following constructions:
AleksandrR [38]

Only you can do that yourself, no one online can do that for you.

8 0
3 years ago
Other questions:
  • How many whole servings of 3/2 cups are in 67/3 cups of fruit punch
    9·1 answer
  • A table titled hours spent studying has entries 4, 5, 9, 15, 8, 2, 20, 1, 6, 9, 10, 8, 10, 8, 1, 2.
    7·2 answers
  • Find a positive angle less than one rotation that is coterminal with –308°.
    10·2 answers
  • Rectangle ABCD is graphed in the coordinate plane. The following are the vertices of the rectangle: A(-4, -2), B(-2, -2), C(-2,
    7·1 answer
  • A fish take contains 16 goldfish and 24 guppies. If you randomly select 2 fish, what is the probability that they are both goldf
    10·1 answer
  • What's the sum 3/6 and 1/16
    6·2 answers
  • The graph of the function f(x) is shown below.
    8·1 answer
  • Drag numbers to fill in the table to organize the information in this problem. Carter works for a fence company. He can paint a
    6·1 answer
  • Factor 140c+28-14a to identify the equivalent expressions.
    14·1 answer
  • The perimeter of a rectangular goat pen is 28 meters. The pen is 5 meters wide. How long is it?
    10·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!