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.

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 `ai` symbol to the right of 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 `ai2` symbol to the right of 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 `12 + 22` evaluates to `5` because `12 + 22` equals `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.

```{
12,
12+22,
12+22+32,
12+22+32+42,
12+22+32+42+52,
12+22+32+42+52+62,
12+22+32+42+52+62+72
}
```

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.