# Finding Shortcuts to Solve Big Math Problems

• slug
In summary, there is a formula to solve a long series of addition of say 18 through 10748, but it is easier to use multiplication.
slug
is there a formula to solve a long series of addition of say 18 through 10748
instead of doing the tedious summing of 18+19+20+...+10748
there probably is but i found a formula yesterday and I want see if it is the same

how about multiplication - is there an easy way of computing 97! (factorial)
instead of using a computer or multiplying 1*2*3*4...*97

There's the old standby $\sum_{i=1}^{n}i=\frac{n(n+1)}{2}$ that will handle your sum problem with little difficulty.

You could use Stirling's series to approximate a factorial.

shmoe said:
There's the old standby $\sum_{i=1}^{n}i=\frac{n(n+1)}{2}$ that will handle your sum problem with little difficulty.

So that, of course, 18+19+20+...+10748= (1+ 2+ ...+ 10748)- (1+ 2+ ...+ 17)= (10749)(10748)/2- (18)(17)/2= (10749)(5374)- (9)17= 57764973.

$$\sum_{i=a}^{n}i=\frac{(n-a+1)(n+a)}{2}$$

will also do

AntonVrba said:
$$\sum_{i=a}^{n}i=\frac{(n-a+1)(n+a)}{2}$$

will also do

Thats the same thing, only slightly simplified.

HallsofIvy said:
So that, of course, 18+19+20+...+10748= (1+ 2+ ...+ 10748)- (1+ 2+ ...+ 17)= (10749)(10748)/2- (18)(17)/2= (10749)(5374)- (9)17= 57764973.

I prefer to think of it in this way:

Please follow my thinking, it is the same as that of Gauss with his SUM(1:100) problem. To solve it you pair off each number, starting with the largest (LA) and smallest (s) andsathen take (L-1) and (S+1) sand so on. With an even number as the largest number, there is always one number that cannot be paired up, once this is found, one subtracts the smallest number from it, to give the number of complete pairs, and multiplies by the sum of L and S. Tsa give the final answer, one then adds the number that cannot be paired.

Take the largest numebr in numberoup (10748) and the smallest (18).

Add them together (10766) , and divide by two (5383). This gives the number that does not have a pair.

Now, subtract 18 from this number (5365). Multiply 5365 by 10766 to give 57759590. Now add on to this 5383, and one gets 57764973

AlgebraiclAlgebraicallyr no miktex but i have never known how to use it in forums).

( { ( (L+S) / 2 ) - S } {L+S}) + { ( L + S ) / 2 } = answer

If one take L+s = T

Then it is much easier to say,
T{(T^2)/2) - ST} + {T/2}

I hope somebody could put that mess into MiXTeX or LaTeX - but i hope my logic comes through.

Regards,

Ben

Last edited by a moderator:
Halls answer is correct. Do you not believe the formula in my post?

"Take the largest numebr in numberoup (10478) and the smallest (18)."

You've transposed the digits 4 and 7, the sum he calculated was from 18 to 10748.

Post corrected

Ben

its not possible simplify factorials is it? it tried but i just end up with long polynomials

## 1. What is the purpose of finding shortcuts in solving big math problems?

The purpose of finding shortcuts in solving big math problems is to save time and effort. By identifying alternative methods or patterns, you can solve complex problems more efficiently and accurately.

## 2. How can I identify shortcuts in solving big math problems?

You can identify shortcuts by looking for patterns or commonalities among similar problems. It also helps to approach the problem from different angles and to break it down into smaller, more manageable parts.

## 3. Are shortcuts always reliable and accurate?

No, shortcuts may not always be reliable and accurate. While they can be helpful in solving certain types of problems, they may not work for every situation. It's important to double check your work and make sure the shortcut is applicable to the specific problem.

## 4. Can I use shortcuts in all areas of math?

Yes, you can use shortcuts in all areas of math, including algebra, geometry, calculus, and more. However, some shortcuts may be more applicable to certain types of problems or concepts.

## 5. How can I practice and improve my ability to use shortcuts in math?

The best way to practice and improve your ability to use shortcuts in math is to solve a variety of problems and actively look for patterns and alternative methods. You can also seek out resources and practice problems specifically geared towards shortcuts and problem-solving strategies.

• Linear and Abstract Algebra
Replies
2
Views
999
• Differential Equations
Replies
1
Views
2K
• Computing and Technology
Replies
24
Views
2K
• Science and Math Textbooks
Replies
28
Views
3K
• General Math
Replies
7
Views
2K
• General Math
Replies
7
Views
1K
• Math Proof Training and Practice
Replies
25
Views
2K
• General Math
Replies
26
Views
3K
• Engineering and Comp Sci Homework Help
Replies
8
Views
1K
• General Math
Replies
4
Views
2K