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Assuming the coefficients are always 1, 4, 1, 4, 1, 4 so on:

\(\dfrac1{3^1} + \dfrac4{3^2} + \dfrac1{3^3} + \dfrac4{3^4} + \cdots\\ = \dfrac13 \left(1 + \dfrac19 + \dfrac1{9^2} + \cdots\right) + \dfrac4{3^2} \left(1 + \dfrac19 + \dfrac1{9^2}+\cdots\right)\\ = \left(\dfrac{1}{3} + \dfrac49\right)\left(\dfrac1{1 - \dfrac19}\right)\\ = \dfrac78\)

sumfor(n, 1, 100,  1/(3^(2*n - 1))  +  4 / 3^(2*n))==It converges to 7/8

\(z^2 + 2 + \dfrac1{z^2} = \left(z + \dfrac1z\right)^2 = 1\\ z^2 + \dfrac1{z^2} = 1 - 2 = -1\)

The sum is equal to 5/3.

The only numbers it can be are: 16 and 61, 25 and 52, and 34 and 43. 

Now, can you tell me which pair of numbers has a difference of 27?

Hint: The correct answer is the smaller number of the correct pair

By Euler's THeorem, 5^{100} = 125 mod 1000 so the sum of the three digits is 1 + 2 + 5 = 8.

The digit in the ones place is three more than the digit in the tens place.

Judging only by this, the number could be 14, 25, 36, 47, 58, 69. There is only one prime number in this list.

\(z^5 = 1\\ z^5 - 1 = 0\\ (z - 1)(z^4 + z^3 + z^2 + z + 1) = 0\\ z^4 + z^3 + z^2 + z + 1 = 0 \text{ ($\because z \neq 1$)}\\ \dfrac{z^4 + z^3 + z^2 + z + 1}{z^2} = 0\\ z + \dfrac1z + z^2 + \dfrac1{z^2} = \boxed{-1}\)

\(\log_2 \left(\dfrac{2}1\right) + \log_2\left(\dfrac32\right)+\cdots + \log_2 \left(\dfrac{64}{63}\right) \\ = \log_2 \left(\dfrac21 \times \dfrac32 \times \cdots \times \dfrac{64}{63}\right)\\ = \log_2 64\\ = 6\)

The answer is obvious at this point.

Solve for z: z (approx) = 0.618017 + 0.28182i

Then z + 1/z + z^2 + 1/z^2 = 0

Let the total amount of money divided among the 3 persons==M

13 + 12 + 7 ==32 sum of all ratios

13 - 7 ==6 - difference in the ratios of Alice and Carl

6/32M == 360, solve for M

M ==$1,920 - total amount to be divided among the 3 persons.

1,920 / 32 ==$60 - share of each person

60  x  13 ==$780 - share of Alice

60  x  12 ==$720 - share of Bob

60  x   7 ==$420  - share of Carl

Check: $780  -  $420 ==$360 Alice's share > than Carl's share.

1m->0,001km and 1sec -> 1/60*60 so 1m/s->0,001/(1/3600) km/h = 3.6 km/h but you have 1.5 m/s so 1.5*3.6=5.4 km/h

Seriously? 25, 25, 70 cannot be the sides of a triangle!

Observation: \((1 + \sqrt 2)^8 + (\sqrt 2-1)^8\) is an integer, and \(0 < (\sqrt 2- 1)^8 < \dfrac12\). That integer is the required integer.

Using binomial theorem,

\(\quad(1 + \sqrt 2)^8 + (\sqrt 2-1)^8\\ =\displaystyle\sum_{k = 0}^8 \binom{8}k ((\sqrt 2)^k + (-\sqrt 2)^k)\\ =\displaystyle\binom{8}0 (2) + \binom82 (2(\sqrt 2)^2) + \binom84 (2(\sqrt 2)^4) + \binom86 (2(\sqrt 2)^6) + \binom88 (2(\sqrt 2)^8)\\ = 1154\)

Note that \(\sqrt 2 - 1 \approx 0.414\), so \(0 < (\sqrt 2 - 1)^8 < \sqrt 2 - 1 < \dfrac12\). That means \((1 + \sqrt 2)^8\) plus a real number way smaller than 1/2 gives 1154.

Therefore, the nearest integer to \((1 + \sqrt 2)^8\) is 1154.

Remarks: The error is \(\left|(1 + \sqrt2)^8 - 1154\right| = (\sqrt 2 - 1)^8 \approx 8.665517772 \times 10^{-4}\), which is really really small.

A similar problem is solved here:
https://web2.0calc.com/questions/inverse-mod#r4

Similar solved problem here, with only the dilation factor changed:

https://web2.0calc.com/questions/help-please_79855

Let x be the number. \(\dfrac{5x}{x + 7} = 8\)

Multiplying by (x + 7) on both sides gives 

\(5x = 8(x + 7)\\ 5x = 8x + 56\)

Now this is just a linear equation. Can you try to solve it?

Let r% be such rate of interest.

\(P\displaystyle\lim_{n\to\infty}\left(1 + \dfrac{r\%}n\right)^{3n} = 2P\\ e^{3r\%} = 2\\ 3r\% = \ln(2)\\ r\% = \dfrac{\ln(2)}3 \approx 0.231\)

The rate must be at least 23.1%.

2 x $10 ==$20 - cost before 1/4 increase.

$20 x 1.25 ==$25 - cost after 1/4 increase

$20 + $25 ==$45.00 - total cost of 4 flash drives.

\(\quad\displaystyle \sqrt{2}+\frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{2}} \\= \sqrt 2 + \sqrt 3 + \dfrac 2{\sqrt 2} \\= 2 \sqrt 2 + \sqrt3\)

Now, the values of a, b, and c are obvious.

Ln(2) / 3 ==23.105% compounded continuously will double your money in 3 years.

How about use 24(nCr)2 =276+ 24squares(31^2, 37^2, 41^2, 43^2, 47^2,.....and so on to 139^2

The ratio can be expressed as 9x:2x:5x. we get the equation:

\(9x*2+2x*5+5x*10-9*\frac{4}{9}x*2-50=300\)

Simplifying it we get:

 \(18x+10x+50x-8x=350\)

\(70x=350\)

Soving it we get x=5. This means that there are \(5*5=\boxed{25}\) $10 notes

can i ask how you got this answer?