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nordsb [41]
3 years ago
12

Find the sum of the digits of the number 6+66+666+6666 + ... +666...66, where the last number contains 100 digits.

Mathematics
1 answer:
Contact [7]3 years ago
4 0

Answer:

<em>Sum of the digits of the number </em>

     =  ((\frac{60((10)^{100}-1) }{81}) - \frac{600}{9} )

Step-by-step explanation:

<u><em>Step(i):</em></u><em>-</em>

<em>Given series </em>

<em>6+66+666+6666 + ... +666...66 up to 100 digits</em>

<em>Taking common '6'</em>

6 ( 1 + 11 +111+ 1111+1111+.................11111....11 100 digits)

<em>Multiply '9' and divisible by'9'</em>

\frac{6}{9} X9( 1+11 +111 +1111 +1111+ ..........11111..up to .100 digits )

<em>Multiply inside '9' </em>

\frac{6}{9}( 9+99 +999 +9999 +.....+ ..........9999..up to .100 digits )

\frac{6}{9}( (10-1)+(100-1)+(1000-1) +(10,000-1)+.....+ ............up to .100 digits )

\frac{6}{9}( (10)+(100)+(1000) +(10,000)+.....+ ............up to .100 digits ) - ( 1+1+1+1+........up to 100 digits)

<u><em>Step(ii):-</em></u>

<u><em>we know that sum of geometric series </em></u>

<u><em></em></u>S_{n} = \frac{a(r^{n}-1) }{r-1}<u><em></em></u>

<u><em>we know that </em></u>

<em>a + a + a+........n terms = n a</em>

<em></em>\frac{6}{9}( (10)+(10)^{2} +(10)^{3}  +(10)^4+.....+ ............up to .100 digits  - (100(1))<em>  ....(i)</em>

<em>The sum of the 100 digits in geometric series</em>

<em></em>S_{100} = \frac{10((10)^{100}-1) }{10-1}<em></em>

<em>Now the equation (i)</em>

<em> The sum of the digits of the number  </em>

<em>       </em>= \frac{6}{9}  ((\frac{10((10)^{100}-1) }{10-1}) - 100)<em></em>

<u><em>Final answer </em></u><em>:-</em>

Sum of the digits of the number

     =  ((\frac{60((10)^{100}-1) }{81}) - \frac{600}{9} )

<em></em>

<em></em>

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Answer:

a. 5 represents the number of calories that come from other ingredients.

b. Neither 8 nor 3 satisfy the given equation, so neither 8 nor 3 can be the solution to the equation.

c. <em>c = 5.5</em>

Step-by-step explanation:

We are given the equation:

4c + 5 = 27

Where c represents the grams of carbohydrates

27 is the total number of calories that are received by the granola bite.

4 is the number of calories from 1 gram of carbohydrates and

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Answer of Part a:

5 is the number of calories that are received from other ingredients.

Part b:

Let us take left hand side (LHS) and put values of c = 8 and 3 one by one.

So, LHS becomes:

4 \times 8 + 5 = 37 \neq RHS\ i.e.\ 27

4 \times 3 + 5 = 17 \neq RHS\ i.e.\ 27

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Part c:

4c + 5 = 27\\\Rightarrow 4c = 22\\\Rightarrow \bold{c = 5.5}

So, solution to the equation is: <em>c = 5.5</em>

8 0
3 years ago
Of all of the individuals who develop a certain rash, suppose the mean recovery time for individuals who do not use any form of
VashaNatasha [74]

Answer:

z(s) is in the rejection  region. We reject  H₀. We dont have enought evidence to support that the cream has effect over the recovery time

Step-by-step explanation:

Sample information:

Size     n = 100

mean   x = 28,5

Population information

μ₀  =  30

Standard deviation  σ   = 8

Test Hypothesis

Null Hypothesis          H₀                x =  μ₀

Alternative Hypothesis   Hₐ           x <  μ₀

We assume CI = 95 %   then α = 5 %    α = 0,05

As the alternative hypothesis suggest we should develop a one tail-test on the left ( we need to find out if the cream have any effect on the rash), effects on the rash could be measured as days of recovery

A z(c) for 0,05 from z-table is:  z(c) = - 1,64

z(s) = ( x - μ₀ ) / σ/√n

z(s) = ( 28,5 - 30 ) / 8/√100

z(s) = - 1,5 * 10 / 8

z(s) =  - 1,875

Comparing z(s)  and z(c)

|z(s)| < |z(c)|        1,875  > 1,64

z(s) is in the rejection  region. We reject  H₀. We dont have enought evidence to support that the cream has effect over the recovery time

3 0
3 years ago
Find the difference of(ab+8a+1)-(-6ab+4)
vagabundo [1.1K]
Hey there!

When finding the difference of expressions like these, there's a few things you should keep in mind. 

1. When finding the difference of two polynomials, multiply the polynomial that implied to be "negative" by –1 to expand the problem and make it easier to solve. 

2. You combine terms based on their unknown terms, like x, y, x², ab, etc. You cannot combine the terms x and x², but you can combine the terms 6b and 8b by adding them. 

3. You can rearrange your terms however you see fit. If a polynomial is implied as "positive", you can multiply that entire polynomial by +1 to get rid of the parentheses. 

With all that in mind, you can go ahead and solve for your difference, like so:

(ab+8a+1)-(-6ab+4)

(ab+8a+1) (-1(-6ab+4))

(ab+8a+1) + (6ab-4)

ab+6ab+8a+1-4

7ab+8a-3

7ab+8a-3 will be your difference. 

Hope this helped you out! :-)
8 0
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