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LESSON 1: ARITHMETIC SEQUENCES

Arithmetic Sequences:

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 , t3 , 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   tn = a + (n – 1)d

·        Arithmetic sequences are linear functions with domain the natural numbers N = {1, 2, 3, 4, 5, …} Example 1:

For the arithmetic sequence above, find  t7, t11 and the general term tn.

Solution:

 a = -2 d = 5 n = 7, 11, n Example 2: Finding specific terms and the General (nth) Term.

Given the sequence –3, -7, -11, -15, …

a) Show that the sequence is arithmetic.

b) Find  t7, t11 and the general term tn.

Solution: a = - 3 d = - 4 n = 7, 11, n Example 3: Finding the Number of Terms in a Given Sequence.

Given the sequence 4, 1, -2, …, -65.  Find the number of terms in the sequence.

Solution:

Let the last term be tn.

 a = 4 d = - 3 n = ? tn = - 65 Example 4: Solving a Sequence given two terms.

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

Solution: 