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

Contact [7]

Answer:

Sum of the digits of the number

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

Step-by-step explanation:

Step(i):-

Given series

6+66+666+6666 + ... +666...66 up to 100 digits

Taking common '6'

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

Multiply '9' and divisible by'9'

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

Multiply inside '9'

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

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

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

Step(ii):-

we know that sum of geometric series

$S_{n} = \frac{a(r^{n}-1) }{r-1}$

we know that

a + a + a+........n terms = n a

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

The sum of the 100 digits in geometric series

$S_{100} = \frac{10((10)^{100}-1) }{10-1}$

Now the equation (i)

The sum of the digits of the number

$= \frac{6}{9} ((\frac{10((10)^{100}-1) }{10-1}) - 100)$

Final answer :-

Sum of the digits of the number

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

4 0

3 years ago

Adrian Mole is 13 and ¾ years old. His dad is 81/4 years older than him and his mum is 9/6 of a year younger than his dad. How o

zhenek [66]

Dad is 34, mom is 32 and 1/2

6 0

3 years ago

What is the slope of the line passing through the point 3,5 and -1,7

creativ13 [48]

Answer:

m = -1/2

Step-by-step explanation:

As we move from the point (-1, 7) to the point (3, 5), x increases by 4 (this is the "run") and y decreases by 2 (this is the "rise").

Thus, the slope of the line connecting these two points is

m = rise / run = -2/4, or m = -1/2.

5 0

3 years ago

F 7x = 42, find the value of 44 - 8x

Pie

The value is -4.
7x=42
/7 /7 (Divide both sides by 7)
x=6
If we insert x=6 into 44-8x, it is 44-8(6), which is 44-48.
44-48=-4, so the answer is -4.

3 0

2 years ago

Kali has a 6 meter ladder that she wants to use to get her cat out of a tree. She puts the base of the ladder 2

Mademuasel [1]

Answer:

5.7m

Step-by-step explanation:

the easiest way to solve for a side is to draw the question in a diagram (i have done this below). As it is a right-angled triangle and we know 2 of the side, we can use the Pythagoras theorem:

$a^{2} +b^{2} =c^{2}$

sides a and b are the two shorter sides.

side c is the hypotenuse (longest side)

as shown in the picture below, because we know both sides c and b, we need to change the equation around:

$c^{2} -b^{2} =a^{2}$

replace these sides with their numbers:

$6^{2} -2^{2} = 32$ (remember that 32 equals side a squared therefore we need to square root 32 to find the length of side a:

$\sqrt{32} =5.65685...$

the question says to round to the nearest tenth meter:

Missing side = 5.7m

I hope this was helpful :-)

8 0

3 years ago