1. The document defines arithmetic means as the terms that are between the first and last terms (called the arithmetic extremes) of a finite arithmetic sequence. 2. It provides a formula for calculating the common difference (d) between terms of an arithmetic sequence when more than two terms are involved. 3. Examples are given of using the average and common difference formulas to calculate arithmetic means between given numbers or to insert additional means into a sequence.

1. ARITHMETIC MEANS
AND 𝒏𝒕𝒉
TERM OF AN
ARITHMETIC SEQUENCE
Week 2

2. What I Need to Know
1.define arithmetic means,
2.determine arithmetic means of a sequence and
3.solves problems involving 𝑛𝑡ℎ term of an arithmetic
sequence.

3. 𝒏𝒕𝒉
Term of an Arithmetic Sequence

8. Computing Arithmetic Means
◦ The first and last terms of a finite arithmetic sequences are
called arithmetic extremes, and the terms in between are
called arithmetic means. In the sequence 4, 8, 12, 16, 20, 24;
the terms 4 and 24 are the arithmetic extremes, while 8, 12, 16,
and 20 are the arithmetic means. Also, 8 is the arithmetic mean
of the arithmetic extremes, 4 and 12.
◦
◦ The arithmetic mean between two numbers is sometimes called
the average of two numbers. If more than one arithmetic means
will be inserted between two arithmetic extremes, the formula
for d, d =
an−ak
n−k
, can be used.

10. Let’s Try!
A.What is the arithmetic mean between 10 and 24?
◦ Solution
a.Using the average formula, get the arithmetic mean of
10 and 24.
◦
a.Thus,
10+24
2
= 17 is the arithmetic mean.

11. A. Insert three arithmetic means between 8 and 16.
◦ Solution:
a. If three arithmetic means will be inserted between 8 and 16, then a1= 8 and a5 = 16.
◦ 8, _____, _____, _____, 16
◦ a1 a2 a3 a4 a5
◦
a. Using the formula for d, compute for the common difference.
◦ d =
an−ak
n−k
◦ =
a5−a1
5−1
◦ =
16−8
5−1
◦ =
8
4
◦ = 2

12. a. The arithmetic means are a2, a3, and a4.
◦ a2 = a1 + d
◦ = 8 + 2
◦ = 10
◦
◦ a3 = a2 + d
◦ = 10 + 2
◦ = 12
◦
◦ a4 = a3 + d
◦ = 12 + 2
◦ = 14
◦
a. Thus, the three arithmetic means between the arithmetic extremes, 8 and 16, are 10, 12, and
14.

13. A. Insert two arithmetic means between 2 and 4 2
◦ Solution:
a. If two arithmetic means will be inserted between 2 and 4 2, then a1= 2 and a4 = 4 2.
◦ 2, _____, _____, 4 2
◦ a1 a2 a3 a4
b. Using the formula for d, compute for the common difference.
◦ d =
an−ak
n−k
◦ =
a4−a1
4−1
◦ =
4 2− 2
4−1
◦ =
3 2
3
◦ = 2

14. a. The arithmetic means are a2 and a3
◦ a2 = a1 + d
◦ = 2 + 2
◦ = 2 2
◦
◦ a3 = a2 + d
◦ = 2 2 + 2
◦ = 3 2
a. Thus, the two arithmetic means between 2 and 4 2 are 2 2
and 3 2.