Sequence and Series: Arithmetic Progression – SS1 Mathematics Lesson Note (First Term)

An arithmetic progression (A.P.) is a sequence in which each term differs from the one before it by a fixed number, called the common difference, d.

• nth term: Un = a + (n – 1)d.
• Sum of the first n terms: Sn = n/2 [2a + (n – 1)d].

Behavioural objectives

By the end of the lesson, students should be able to:

1. find the next terms of a sequence and state the rule that gives them;
2. find any term of a sequence from its nth term;
3. define an arithmetic progression and identify its first term and common difference;
4. use the formula Un = a + (n – 1)d to find a term of an A.P.;
5. find the sum of an arithmetic series using the sum formulas.

What is a sequence?

A sequence is a list of numbers arranged in order, where a rule gives each term.

Finding the rule and the next two terms

What is the general term of a sequence?

The general term, or nth term, is a formula that gives any term from its position n.

• Example: the sequence 1, 5, 9, 13, 17 has nth term 4n – 3.
• n: the position of the term in the list.

Check:

• 5th term = 4(5) – 3 = 20 – 3 = 17
• 10th term = 4(10) – 3 = 40 – 3 = 37

Example: finding terms from the nth term

Find the 6th and 9th terms of the sequence whose nth term is (a) 2n + 1, (b) 3 – 5n.

Solution (a): 2n + 1

• 6th term = 2(6) + 1 = 12 + 1 = 13
• 9th term = 2(9) + 1 = 18 + 1 = 19

Solution (b): 3 – 5n

• 6th term = 3 – 5(6) = 3 – 30 = -27
• 9th term = 3 – 5(9) = 3 – 45 = -42

What is an arithmetic progression?

An arithmetic progression is a sequence whose terms increase or decrease in equal steps.

• Example: 9, 12, 15, 18, 21, … has first term 9 and common difference +3.
• Next three terms: 24, 27, 30.
• Finding d: subtract any term from the term after it, for example 12 – 9 = 3.

What are the kinds of A.P.?

There are two kinds of A.P.: increasing and decreasing.

• Increasing A.P.: d is positive, as in 9, 12, 15, 18 (d = 3).
• Decreasing A.P.: d is negative, as in 18, 12, 6, 0 (d = -6).

What symbols are used in an A.P.?

How do you find the nth term of an A.P.?

The nth term of an A.P. is Un = a + (n – 1)d.

Take the sequence 9, 12, 15, 18, 21. Here a = 9 and d = 12 – 9 = 18 – 15 = 3.

• 1st term: U1 = 9 = a
• 2nd term: U2 = 9 + 3 = a + d
• 3rd term: U3 = 9 + 3 + 3 = a + 2d
• 10th term: U10 = 9 + 9(3) = a + 9d
• nth term: Un = 9 + (n – 1)3 = a + (n – 1)d

The number multiplying d is always one less than the position of the term.

Example 1: finding a term

Given the A.P. 9, 12, 15, 18, …, find the 50th term.

1. Write the values: a = 9, d = 3, n = 50.
2. Write the formula: Un = a + (n – 1)d.
3. Substitute: U50 = 9 + (50 – 1)(3).
4. Simplify: U50 = 9 + 49(3) = 9 + 147.
5. Answer: U50 = 156.

Example 2: finding the first term

The 43rd term of an A.P. is 26 and the common difference is ½. Find (a) the 1st term, (b) the 50th term.

(a) The 1st term

1. Write the values: U43 = 26, d = ½, n = 43, a = ?
2. Substitute: 26 = a + (43 – 1)(½).
3. Simplify: 26 = a + 42(½) = a + 21.
4. Subtract: a = 26 – 21 = 5.

(b) The 50th term

1. Write the values: a = 5, d = ½, n = 50.
2. Substitute: U50 = 5 + (50 – 1)(½).
3. Simplify: U50 = 5 + 49(½) = 5 + 24½.
4. Answer: U50 = 29½.

What is an arithmetic series?

A series is the expression formed when the terms of a sequence are added.

• Example: the sequence 1, 3, 5, 7, 9, 11 gives the series 1 + 3 + 5 + 7 + 9 + 11.
• Arithmetic series: a series whose terms form an A.P.

What are the kinds of series?

There are two kinds of series: finite and infinite.

• Finite series: a series with a last term (the nth term).
• Infinite series: a series whose terms never end, as in 1 + 3 + 5 + 7 + 9 + 11 + …
• Sum of an infinite series: it is often impossible to find, so we sum finite series.

What is the difference between a sequence and a series?

• Sequence: a list of numbers separated by commas, such as 1, 3, 5, 7.
• Series: the same numbers joined by plus signs, such as 1 + 3 + 5 + 7.

How do you find the sum of an A.P.?

The sum of an A.P. is found by pairing the first and last terms: S = n/2 (a + L).

Worked example: adding 1, 3, 5, …, 15

Find the sum of 1, 3, 5, 7, 9, 11, 13, 15. Here n = 8.

1. Write the sum forwards: S = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 (eqn 1).
2. Write the sum backwards: S = 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 (eqn 2).
3. Add the two equations: 2S = 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16.
4. Simplify: 2S = 8(16) = 128.
5. Divide by 2: S = 64.

How is the sum formula derived?

Let L be the last term of a general arithmetic series.

1. Write forwards: S = a + (a + d) + (a + 2d) + … + (L – 2d) + (L – d) + L (eqn 1).
2. Write backwards: S = L + (L – d) + (L – 2d) + … + (a + 2d) + (a + d) + a (eqn 2).
3. Add the two equations. Each pair of terms adds up to (a + L), and there are n pairs.
4. So 2S = n(a + L).
5. Divide by 2: S = n/2 (a + L).
6. Since L = Un = a + (n – 1)d, substitute for L: S = n/2 (a + a + (n – 1)d).
7. Simplify: S = n/2 [2a + (n – 1)d].

Which sum formula should you use?

Example: sum of the first 20 terms

Find the sum of the first 20 terms of the series 16 + 9 + 2 + …

1. Find the values: a = 16, d = 9 – 16 = -7, n = 20.
2. Write the formula: S = n/2 [2a + (n – 1)d].
3. Substitute: S = 20/2 [2(16) + (20 – 1)(-7)].
4. Simplify: S = 10 [32 + 19(-7)] = 10 (32 – 133).
5. Answer: S = 10(-101) = -1010.

Practice questions

1. Find the 4th term of an A.P. whose first term is 2 and common difference is 0.5.
   (a) 4 (b) 4.5 (c) 3.5 (d) 2.5
2. In an A.P. the difference between the 8th and 4th terms is 20, and the 8th term is 1½ times the 4th term. Find the common difference.
   (a) 5 (b) 7 (c) 3 (d) 10
3. Find the first term of the sequence in question 2.
   (a) 70 (b) 45 (c) 25 (d) 5
4. The next term of the sequence 18, 12, 6, 0 is
   (a) 12 (b) 6 (c) -6 (d) -12
5. Find the number of terms of the sequence ½, ¾, 1, …, 5½.
   (a) 21 (b) 4¾ (c) 1 (d) 22
6. (i) Write down the 15th term of the sequence 2/(1 × 3), 3/(2 × 4), 4/(3 × 5), 5/(4 × 6), …
   (ii) An A.P. has 3 as its first term and 4 as its common difference.
   (a) Write an expression in its simplest form for the nth term.
   (b) Find the 10th term.

Answers: 1 (c); 2 (a); 3 (c); 4 (c); 5 (a); 6 (i) 16/255, (ii)(a) 4n – 1, (b) 39.

Evaluation

1. Find the next two terms and the rule for each sequence:
   (a) 1, ½, ¼, ⅛, ____, ____
   (b) 100, 96, 92, 88, ____, ____
   (c) 2, 4, 6, 8, 10, ____, ____
   (d) 1, 4, 9, 16, 25, ____, ____
2. For each set, (i) arrange the numbers in ascending order, (ii) find the next two terms:
   (a) 19, 13, 16, 22, 10
   (b) -2½, 5½, 3½, 1½, -½
3. Find the 15th term of the sequence whose nth term is 3n – 5.
4. Find the 20th term of the A.P. 7, 10, 13, …
5. Find the sum of the first 10 terms of the A.P. 2, 5, 8, …

Assignment

Eight wooden poles are used as pillars, and their lengths form an A.P. The second pole is 2 m long and the 6th pole is 5 m long. Give the lengths of the poles in order and find the sum of their lengths.