Shortcut to calculate

Shortcut to calculate

I have an equation as below:

3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + 3^7 + 3^8 + 3^9 + 3^10

Note: ^ stands for exponent

Anybody knows the shortcut to calculate that equation i/o multiplying and plus one by one. Please advise

ice_breaker

Posts: 6
Joined: Sat Jan 16, 2010 3:40 am

That is geometric series.
Math Tutor

Posts: 313
Joined: Sun Oct 09, 2005 11:37 am

Thank you Teacher.

I did have a look at your given site and some sites thru Google. Finally, I figure out the solution as below:

3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + 3^7 + 3^8 + 3^9 + 3^10

1) Get out a common factor
3(1 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + 3^7 + 3^9)

2) Use the formula [(a^n - b^n)/2] , as a=3, b=1, to calculate the parenthesis, then multiply by 3
3 * [(3^10 - 1^10)/2] = 88 572.

The result is 88,572.

ice_breaker

Posts: 6
Joined: Sat Jan 16, 2010 3:40 am

I do not think that here we can use that formula.
Where you found it? Could I see, please?

The geometric series formula is:
$S = a1 \frac{(1-q^n)}{1-q}$

a1 = 3 - the first number of the sequence
q = 3 - common ratio (because every number is the previous number multiplied by 3)
n=10
so

the sum is
$Sum = 3 \frac{(1-3^{10})}{1-3} = 3 \frac{(1-3^{10})}{1-3}$

$Sum = 3 \frac{(-59048)}{-2} = 88572$
Math Tutor

Posts: 313
Joined: Sun Oct 09, 2005 11:37 am

This formula is provided by a friend of mine. He told me to use that formula to solve this equation.

I think I'd better use yours to standardize my knowledge. By the way, if the equation is like:

1/2 + 1/2^2 + 1/2^3 + 1/2^4 + ..... + 1/2^10

ice_breaker

Posts: 6
Joined: Sat Jan 16, 2010 3:40 am

a1 = 1/2 - the first number of the sequence
q = 1/2 - common ratio
n=10
Math Tutor

Posts: 313
Joined: Sun Oct 09, 2005 11:37 am

Thanks a lot Teacher
ice_breaker

Posts: 6
Joined: Sat Jan 16, 2010 3:40 am

Re: Shortcut to calculate

I do not think that here we can use that formula.
Where you found it? Could I see, please
sajid121

Posts: 10
Joined: Thu Apr 26, 2012 6:04 am

Re: Shortcut to calculate

It is the base formula for sum of geometric series.
Guest

Re: Shortcut to calculate

Hey,
it was very helpful, though I have an exam tomorrow, and the only thing I'm confused is the steps multiplying fractions like:
Image
It would be awesome if someone would show me steps (for beginners), I do know how to cross-multiply, it's just the n's confuse me.
sajid5566

Posts: 1
Joined: Mon May 21, 2012 12:41 am