Logarithms

[Albi]

Guys I'm stuck the dots are confusing me.
I need some help!

 

[Deleted member 4993]

View attachment 22006
Guys I'm stuck the dots are confusing me.
I need some help!

You have (1+ 2 + 3 +.....+ 100) * log(x) = 5050

(1+ 2 + 3 +.....+ 100) is an arithmetic series. Do you know the equation for sum of an arithmetic series?

 

[Albi]

You have (1+ 2 + 3 +.....+ 100) * log(x) = 5050

(1+ 2 + 3 +.....+ 100) is an arithmetic series. Do you know the equation for sum of an arithmetic series?

It is n/2[2*a1 +(n-1)d]

 

[Deleted member 4993]

It is n/2[2*a1 +(n-1)d]

For your problem what are the numerical values of :

a₁= ?

n = ? and

d = ?

 

[Albi]

For your problem what are the numerical values of :

a₁= ?

n = ? and

d = ?

a1=1
d=1
But I don't know what n is

 

[JeffM]

The dots mean all the ones in between. so if I count all the numbers from 1 to 100, how many numbers is that?

 

[pka]

Guys I'm stuck the dots are confusing me.I need some help!

Do you know that the sum of the first \(n\) positive integers is \(\large\dfrac{n(n+1)}{2}~?\)
Here \(n=100\).

 

[Albi]

The dots mean all the ones in between. so if I count all the numbers from 1 to 100, how many numbers is that?

A hundred

 

[JeffM]

A hundred

So n = 100

And now you know what ... means

 

[HallsofIvy]

There is a story that when Gauss was a child, the teacher, just to keep the class busy, made them add all integers from 1 to 100 (or 1000). Gauss wrote a single number on his paper.

He had thought of the sum as
S= 1+ 2+ 3+ ...+ 98+ 99+ 100 and then
S=100+ 99+ 97+ ...+ 3+ 2+ 1 so
2S= 101+ 101+ 101+ ...+ 101+ 101+ 101

2S is 101 100 times so 2S= 10100 and S= 5050.

Just as easily adding 1 to 1000, 2S is 1001 1000 times so 2S= 1001000 and
S= 500500.

For general "n"
S= 1+ 2+ 3+ ...+ (n-2)+ (n-1)+ n
S= n+ (n-1)+ (n- 2)+ ...+ 3+ 2+ 1 so
2S= (n+ 1)+ (n+ 1)+ (n+ 1)+ ...+ (n+1)+ (n+1)+ (n+1)
2S= n+1 n times so 2S= n(n+1) and \(\displaystyle S= \frac{n(n+1)}{2}\).