# Numbers Game: Sigma Summation and Factorial Help

## Level Four

### Learn to Play the Numbers Game Level Four

Select the Play

button at any time to
start or restart the game. Play

resets
your score for this level.
The game's over when either the timer runs
out or you've answered every question.

### Post Your Score

When the game's over
enter a user name and password.
Select the Post Score

button
to see how your score compares to others online.
Each level starts over if you return and
tap the Play

button.
To accumulate the highest score complete a level,
then start another level.

### Game Play

Questions display below the landscape.
For each correct answer your score increases
by forty points.
Tap a digit in the background, then select a *Sigma Summation Σ* or *factorial !* symbol.
Sum a series of integers, sum a series of integers squared, or find the product of a series of integers.

### Sigma Σ Sum Integers

The graphic to the left represents *summation of a series of numbers* or *Sigma Σ*.
Summation of a series of numbers involves simple addition.
Start with the lower limit. End with the upper limit.
Add every digit from the lower to the upper limit.

The `a`

symbol to the right of _{i}*Sigma Σ*
indicates each number as it's summed.
The *n* which displays above the *Sigma Σ* symbol
represents the *upper limit* or ending number.
The *i* below the *Sigma Σ* symbol
represents the *lower limit* or starting number.
Notice *i* equals one.
The lower limit for this particular
icon always equals one. However players select the upper limit.

To form an expression with *Sigma Σ* select a digit
for the upper limit, then select the *Sigma Σ* symbol.
The digit you select substitutes for *n*.

For example select the number `3`

then the *Sigma Σ* symbol.
The upper limit equals `3`

. The lower limit equals `1`

.
The expression equals `1 + 2 + 3`

. The result equals `6`

because `1 + 2 + 3 = 6`

.

Assume the following question displays below the landscape:

Form an expression which evaluates to 10.

To find the correct answer
select `4`

, then tap the *Sigma Σ*
symbol.
The expression
`1 + 2 + 3 + 4`

evaluates to `10`

because `1 + 2 + 3 + 4 = 10`

.
Your score increases by forty points.
However other correct expressions might exist.

### Sigma Σ Sum Numbers Squared

The graphic to the left represents *Σ Squared* or *Sigma Squared*.
The `a`

symbol to the right of _{i}^{2}*Σ Squared*
represents each number as it's squared and summed.
To calculate the result of *Σ Squared* start with the lower limit. End with the upper limit.
Square then add every digit from the lower to the upper limit.
Again *i* equals one. The lower limit for this particular
icon always equals one. The player selects a digit which substitutes for the `n`

and becomes the upper limit.

Assume the following question displays below the landscape.

Form an expression which evaluates to 5.

To find the correct answer
select `2`

, then tap the *Σ Squared*
symbol.
The expression
`1`

evaluates to ^{2} + 2^{2}`5`

because `1`

equals ^{2} + 2^{2}`1 + 4`

and `1 + 4 = 5`

.
Your score increases by forty points.
However other correct expressions might exist.

### n! Factorial Symbol

The factorial symbol *multiplies* a series of numbers.
For example the factorial of `3`

equals `6`

because `1 * 2 * 3 = 6`

.

Select a digit, then select the factorial *n!* symbol.
The digit you select substitutes for *n*.

Assume the following question displays below the landscape.

Form an expression which evaluates to 2.

To find the correct answer
select `2`

, then tap the *n!*
symbol.
The expression
`1 * 2`

evaluates to `2`

because `1 * 2 = 2`

.
Your score increases by forty points.
However other correct expressions might exist in the game.

The factorial symbol provides very rapid increases in value.
For example the factorial of `7`

equals `5040`

.

### Zoom In or Zoom Out

Select the magnifying glass icons below the cloudscape to zoom in and out.

## Summation and Factorial Cheet Sheets

### Cheat Sheet for Summation

The domain for the Numbers Game

is `{1,2,3,4,5,6,7}`

.
The range for `Sigma Σ sum of numbers`

follows.

{ 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5, 1+2+3+4+5+6, 1+2+3+4+5+6+7 }

The result equals `{1,3,6,10,15,21,28}`

.
Therefore with this level of the Numbers Game

,
`Sigma Σ sum of numbers`

calculates to
one of seven values from the set `{1,3,6,10,15,21,28}`

.

Also the formula `Σn = [n x (n + 1)]/2`

results in the sum from 1 to `n`

integers.
For example assume n = 6.
`[6 x (6 + 1)]/2`

equals `[6 x 7]/2`

equals `42/2`

equals `21`

.
Notice the sixth entry in the set `{1,3,6,10,15,21,28}`

also equals `21`

.

### Cheat Sheet for Sum of Squares

The domain for the Numbers Game

level four
is `{1,2,3,4,5,6,7}`

.
The range for `Sigma Σ Squared`

follows.

{ 1^{2}, 1^{2}+2^{2}, 1^{2}+2^{2}+3^{2}, 1^{2}+2^{2}+3^{2}+4^{2}, 1^{2}+2^{2}+3^{2}+4^{2}+5^{2}, 1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}, 1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}+7^{2}}

The result equals `{1,5,14,30,55,91,140}`

.
Therefore `Sigma Σ Squared`

calculates to
one of seven values from the set `{1,5,14,30,55,91,140}`

,
in level four of the Numbers Game

.

### Cheat Sheet for Factorial

The domain for the Numbers Game

level four
is `{1,2,3,4,5,6,7}`

.
The range for `factorial`

*n!* follows.

{ 1x1, 1x2, 1x2x3, 1x2x3x4, 1x2x3x4x5, 1x2x3x4x5x6, 1x2x3x4x5x6x7 }

The result equals `{1,2,6,24,120,720,5040}`

.
Therefore `factorial`

*n!* calculates to
one of seven values from the set `{1,2,6,24,120,720,5040}`

,
in level four of the Numbers Game

.