**High School Number Sense Lesson 76: Adding Fractions in the Form 1/n(n+1) + 1/(n+1)(n+2) + ...**

Trust me--you're going to like today's lesson. It demonstrates another example when math is beautiful--when the "number sense" method of solving it seems a little TOO easy. This concept appeared

**5 times**last year, with a median placement at

**question # 77**.

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It will probably be easier to show examples of today's concept than to explain what you are looking for. Here's one:

**1/6 + 1/12 + 1/20 + 1/30**

**all the same**. Now pay close attention to the denominators. They can be rewritten like this:

**1/(2x3) + 1/(3x4) + 1/(4x5) + 1/(5x6)**

**How to Solve:**

- Mentally rewrite each fraction with 2 factors in the denominator (one smaller times one larger), ensuring that the larger factor in the first fraction becomes the smaller factor in the 2nd fraction, and so on.
- Add up the numerators to get your numerator.
- Multiply the
**far left**denominator factor (the smaller factor in the first fraction) by the**far right**denominator factor (the larger factor in the last fraction) to get your denominator. - Reduce if possible.

So in the above example:

- Numerator: 1 + 1 + 1 + 1 =
**4** - Denominator: 2 x 6 =
**12** - Answer: 4/12 =
**1/3**

One of the really neat things about this concept is that the factors in the denominators don't have to be consecutive integers. They could be consecutive odd integers, or consecutive even integers, or even some other pattern. Also, the numerators don't always have to be 1; they just need to be the same in each fraction.

**Example 1: 1/30 + 1/42 + 1/56 = ___**

- Think of this as
**1/(5x6) + 1/(6x7) + 1/(7x8)** - Numerator: 1 + 1 + 1 =
**3** - Denominator: 5 x 8 =
**40** - Answer:
**3/40**, which does not reduce

**Example 2: 1/30 + 1/42 + 1/56 + 1/72 = ___**

- Think of this as
**1/(5x6) + 1/(6x7) + 1/(7x8) + 1/(8x9)** - Numerator: 1 + 1 + 1 + 1 =
**4** - Denominator: 5 x 9 =
**45** - Answer:
**4/45**, which does not reduce

**Example 3: 1/3 + 1/15 + 1/35 = ___**

- Think of this as
**1/(1x3) + 1/(3x5) + 1/(5x7)** - Numerator: 1 + 1 + 1 =
**3** - Denominator: 1 x 7 =
**7** - Answer:
**3/7**, which does not reduce

**Example 4: 1/8 + 1/24 + 1/48 + 1/80 = ___**

- Think of this as
**1/(2x4) + 1/(4x6) + 1/(6x8) + 1/(8x10)** - Numerator: 1 + 1 + 1 + 1 =
**4** - Denominator: 2 x 10 =
**20** - Answer: 4/20 =
**1/5**

**Example 5: 1/15 + 1/30 + 1/48 + 1/80**

- Think of this as
**1/(3x5) + 1/(5x6) + 1/(6x8) + 1/(8x10)**. Notice that the factors aren't evenly spaced. - Numerator: 1 + 1 + 1 + 1 =
**4** - Denominator: 3 x 10 =
**30** - Answer: 4/30 =
**2/15**

**Example 6: 5/24 + 5/48 + 5/80 + 5/120**

- Think of this as
**5/(4x6) + 5/(6x8) + 5/(8x10) + 5/(10x12)** - Numerator: 5 + 5 + 5 + 5 =
**20** - Denominator: 4 x 12 =
**48** - Answer: 20/48 =
**5/12**

**Here's a free worksheet to help you practice AddF1/n(n+1):**

addf1n_n_1_.pdf |

**Up Next for High School: AddSeqCube**