M3L2 ARITHMETICSEQUENCES & SERIES

By M Willatt

Pictures and summation made on Microsoft Word

We’re going to work with these 4 arithmetic sequences in

order to practice the steps you need to be able to do for M3L2!

Study them for a minute before you jump in…

1. -5, -7, -9, -11, -13, …

2. The first term is 2 and the fifth term is 70.

3. 0, 1.5, 3, 4.5, 6, …

4. The first term is 11 and the third term is -3.

Find the common difference for each arithmetic sequence:

1. -5, -7, -9, -11, -13, … d = term – preceding term = (-7) – (-5) = -

2

2. The first term is 2 and the fifth term is 70.

d = difference between terms = 70-2 = 68 = 17

difference in term #’s 5-1 4

3. 0, 1.5, 3, 4.5, 6, … d = term – preceding term = 1.5 – 0 = 1.5

4. The first term is 11 and the third term is -3.

d = difference between terms = -3 -11 = -14 = -7

difference in term #’s 3-1 2

Reminder on how to write the explicit form of an…

Write the explicit formula for each using the d we just found:

1. -5, -7, -9, -11, -13, … a1= -5 & d = -2 an= -5 + (n-1)(-2)

2. The first term is 2 and the fifth term is 70.

a1= 2 & d = 17 an= 2 + (n-1)17

3. 0, 1.5, 3, 4.5, 6, … a1= 0 & d = 1.5 an= 0 + (n-1)1.5

4. The first term is 11 and the third term is -3.

a1= 11 & d = -7 an= 11 + (n-1)(-7)

Find the 13th term using the explicit formula we just found:

1. an= -5 + (n-1)(-2) a13= -5 + (13-1)(-2)= -5 +(12)(-2) = -29

2. an= 2 + (n-1)17 a13= 2 + (13-1)17= 2 +(12)17 = 206

3. an= 0 + (n-1)1.5 a13= 0 + (13-1)1.5= 0 +(12)1.5 = 18

4. an= 11 + (n-1)(-7) a13= 11 + (13-1)(-7)= 11 +(12)(-7) =-73

Remember a series is the sum of the terms in a sequence.

To add up the 1st “n” terms, we could write:

But what if “n” was 50?

You would have to find the first 50 terms & then add

them.

That’s a lot of work…

Fortunately, there is a formula that only requires

the 1st & last term of the series:

Find the sum of the first 13 terms:

1. a1= -5 a13= -29 𝑆13 =13

2−5 + −29 = -221

2. a1= 2 a13= 206 𝑆13 =13

22 + 206 = 1352

3. a1= 0 a13= 18 𝑆13 =13

20 + 18 = 117

4. a1= 11 a13= -73 𝑆13 =13

211 + −73 = -403

• If you’re asked for the common difference, see what number

is being added between each term. You can subtract 2nd

term – 1st term OR 3rd term – 2nd term, etc.

• If you’re asked for the 50th term, you need to substitute a 50

into the explicit formula for n. If you don’t have the

formula, then write it using a1 and d.

• If you’re asked to find the partial series, you can use the

formula Sn. However, you may need to find the term an first

using the explicit formula.