This document discusses arithmetic sequences and series. It provides 4 example sequences and walks through finding the common difference, writing the explicit formula, and calculating specific terms like the 13th term. It also introduces the formula for finding the sum of the first n terms of a series. Key steps covered include subtracting consecutive terms to find the common difference, writing the explicit formula using the first term and common difference, and using the formula for the partial sum to add the first 13 terms instead of individually.

1. M3L2 ARITHMETIC
SEQUENCES & SERIES
By M Willatt
Pictures and summation made on Microsoft Word

2. 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.

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

4. Reminder on how to write the explicit form of an…

5. 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)

6. 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

7. 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…

8. Fortunately, there is a formula that only requires
the 1st & last term of the series:

9. 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
2
2 + 206 = 1352
3. a1= 0 a13= 18 𝑆13 =
13
2
0 + 18 = 117
4. a1= 11 a13= -73 𝑆13 =
13
2
11 + −73 = -403

10. • 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.