LESSON 1

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UNIT 10 : SEQUENCES AND SERIES

LESSON 3: ARITHMETIC SERIES

Arithmetic Series:

Recall a sequence such as –2, 3, 8, 13, … is called an Arithmetic Sequence. These sequences have the following properties.

· Terms are denoted as t₁ , t₂ , t_{3 ,}referring to term1, term 2, term 3 …

· The difference between successive terms is constant. ie t₂ – t₁ = t₃ – t₂ = t₄ – t₃ etc

· This difference is called the common difference and denoted using the letter d. Here d = 5.

· The first term is denoted using the letter a. Here a = -2.

· Successive terms are found by adding the common difference, d, to the preceding term. Hence t₅ = 13 + 5 = 18 etc.

· The formula for the general term or nth term is t_n= a + (n – 1)d

Text Box: a = t1 = first term
d = common difference
n = term number
tn = nth term or general term

Definition: The sum of the terms of an arithmetic sequence is an Arithmetic Series.

Text Box:

Text Box: n = the number of terms
t1 = a = the first term
tn = l = the last term
d = the common difference
Sn = the sum of the series

Example 1: Given the first few terms.

For the arithmetic series below, above, find t₁₁, the general term t_n and the sum of 40 terms S_40.

Solution:

a = 1

d = 3

n = 11

t_n = ?

S₄₀ = ?

Example 2: Given the first and last terms.

a = - 3

d = - 4

t_n = - 219

n = ?

S_n = ?

Solution:

Example 3: Given two terms.

The fourth and seventh terms of an arithmetic series are 8 and 17 respectively. Find a, d and S₂₆.

Solution: