Algebraic Equation n times...

[lingping7]

The question:
Some girls who were sitting in straight rows, took a speed typing test for typing a certain line as many times as they could. After the test, each girl discovered that the girl on the left of her typed twice as much as she did.
If there were 635 lines in total, find the number of girls.

This problem seems very tricky for me.
I have arrived at an equation: x+2x+4x+8x+16x.......n times, so n is the number of girls.

Please guide me how to solve this problem.

Thanks in advance.

 

[Deleted member 4993]

The question:
Some girls who were sitting in straight rows, took a speed typing test for typing a certain line as many times as they could. After the test, each girl discovered that the girl on the left of her typed twice as much as she did.
If there were 635 lines in total, find the number of girls.

This problem seems very tricky for me.
I have arrived at an equation: x+2x+4x+8x+16x.......n times, so n is the number of girls.

Please guide me how to solve this problem.

Thanks in advance.

Do you know about geometric series?

What is 'x' in your expression: x+2x+4x+8x+16x.......n times?

 

[soroban]

Hello, lingping7!

You are off to a good start . . .

Some girls, who were sitting in a straight row, took a speed typing test
for typing a certain line as many times as they could.
After the test, each girl discovered that the girl on the left of her typed twice as much as she did.
If there were 635 lines in total, find the number of girls.

This problem seems very tricky for me.

I have arrived at an equation: x+2x+4x+8x+16x.......n times, so n is the number of girls.

You have: .\(\displaystyle x + 2x + 2^2x + 2^3x + 2^4x + \cdots + 2^{n-1}x \;=\;635\)

The left side is a geometric series
with first term \(\displaystyle a = x\), common ratio \(\displaystyle r = 2\), and \(\displaystyle n\) terms.

The formula for its sum is: .\(\displaystyle S \:=\:a\dfrac{r^n-1}{r-1}\)
We have: .\(\displaystyle x\dfrac{2^n-1}{2-1} \:=\:635 \quad\Rightarrow\quad x(2^n-1) \:=\:635\)

We note that: \(\displaystyle 635 \,=\,5\cdot127\)

So we have: .\(\displaystyle x(2^n-1) \:=\:5\cdot 127\)

Hence: .\(\displaystyle \begin{Bmatrix}x \:=\: 5 \\ 2^n-1 \:=\:127 & \Rightarrow & 2^n \:=\:128 & \Rightarrow & n = 7 \end{Bmatrix}\)

Therefore, there were 7 girls.
. . The first girl typed 5 lines.