Arithmetic Sequence

Objectives:At the end of the study, the

student must be able to:• define arithmetic sequence;• know if a sequence is an

arithmetic progression;• apply arithmetic sequence in

problems;• insert arithmetic means between

two numbers, and;• get the common difference, first

term, and nth term.

What is it?• A sequence is a set of numbers in a specific

order. What this means is that the set of numbers can be put into a one-to-one correspondence with the Counting Numbers (1, 2, 3, 4, ... ). Thus, you can talk about the 1st element (or term) in a sequence or the 10th element in a sequence or the 101st element in a sequence.

• An arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same, i.e., the difference is a constant.

• It is a sequence in which the difference between two successive terms has a constant (d) which is called the common difference.

General Formula:

tn=a+(n-1)da = first termn = no. of termsd = common difference(term-previous term)

tn = nth term

Points to Ponder• The sequence that begins 1, 4, 7, 10,

13, 16, . . . is an arithmetic sequence since the difference between consecutive terms is always 3.

• The sequence that begins 8, 6, 4, 2, 0, -2, -4, . . . is an arithmetic sequence since the difference between consecutive terms is always -2.

• In order to identify if a pattern is an arithmetic sequence you must examine consecutive terms. If all consecutive terms have a common difference you can conclude that the sequence is arithmetic.

Find the variables being asked.1. 2,7,12 find d and t12

d=5t12=a+(n-1)d

=2+(11-1)5 =2+11(5) =2+55

answer: 572. 6, 10, 14 find t50

d=40 t50=6+(50-1)4 =6+(49)4 =6+196 =202

Examples:

3. Find the 15th term of the A.P if the fifth term is 12 and the tenth term is -3

t5= a + (5-1)d

t10= a + (10-1)d (a+4d=12)-1 a+4d=12 a+9d=-3 a+4(-3)=12 -a-4d=-12 a-12=12 a+9d=-3 a=24

5d=-15 t15= a +(15-1)d d=-3 =24+14(-3)

=14+(-42) = -15

4. Find the value of k if 6-2k, 3k+1, and 5k form an A.P.

(3k+1)-(6-2k)=(5k)-(3k+1)3k+1-6-2k=5k-3k-1

5k-5=2k-13k=4

k= 4/3

5. Insert three arithmetic means between 4 and 10

4, t2,t3,t4,10 t2=4+3/2= 5 ½

t5=a +(5-1)d t3=5 ½+3/2=7

10=4+4d t4=7+3/2= 8 ½

6=4dd=3/2

ACTIVITIES

Good Luck!!!

I. Determine which of the following sequences are in A.P. For those that are in A.P, give the common difference, and the next three terms of the sequence.

1. 0.1, 0.01, 0.001…2. 40, 42, 44, 46…3. 5, 8, 11, 14…4. 1/3, 1/4, 1/6, 1/12…5. 1.2, 1.8, 2.4…6. -11, -7, -3, 1…7. x+2, 2x+1, 3x…8. 1/3, 1, 5/3..9. 5/3, 15/4, 5…10. √2, √3, √4, √5…

II. Given the first term (a), and the common difference (d) of an A.P, find the next 5 terms.

1. a = 2/5 d = 1/102. a = 1.5 d = 0.33. a = 3 d = -54. a = -3 d = 25. a = x+4 d = x-2

III. Find the common difference and insert four arithmetic means between the given numbers.

1. 9 and 242. -25 and 33. 4 and 1794. 50.1 and 50. 75. a and a+126. x + 2 and x + 10

IV. Problem solving. Find the variable being asked.

1. If 5x – 3, x + 2 and 3x – 11 form an A.P, find x and t21.

2. If the first term is -4, and the common difference is 3, what term is 116?

3. The ninth term of an A.P is 15, and the 17th term is 27, find the a and d.

4. The third term of an A.P is 9 and its 7th term is 49, what is the 11th term?

5. A carpenter made a ladder with 16 rungs. The bottom rung is 70 cm. if each succeeding rung is 1 cm shorter than the preceding, how long is the top most rung?

Answers

Test I answers:1. Not A.P2. A.P, d = 2, Next 3 terms = 48, 50, 523. A.P, d = 3, Next 3 terms = 17, 20, 234. A.P, d = -1/12, Next 3 terms = 0, -1/12,

-1/65. A.P, d = 0.6, Next 3 terms = 3, 3.6, 4.26. A.P, d = 4, Next 3 terms = 5, 9, 137. A.P, d = x-1, Next 3 terms = 4x-1, 5x-

2, 6x-38. A.P, d = 2/3, Next 3 terms = 2 1/3, 3, 3

2/39. Not A.P10.Not A.P

Test II answers:1.1/2, 3/5, 7/10, 4/5, 9/102.1.8, 2.1, 2.4, 2.7, 33.-2, -7, -12, -17, -224.-1, 1, 3, 5, 75.2x+2, 3x, 4x-2, 5x-4,

6x-6

Test III answers:1. d = 3, Four arithmetic means = 12,

15, 18, 212. d = 28/5, Four arithmetic means =

-19 2/5, -13 4/5, -8 1/5, -2 3/53. d = 35, Four arithmetic means =

39, 74, 109, 1444. d = 0.12, Four arithmetic means =

50.22, 50.34, 50.46, 50.585. d = 12/5, Four arithmetic means =

a + 12/5, a + 24/5, a + 36/5, a + 48/5

6. d = 8/5, Four arithmetic means = x + 18/5, x + 26/5, x + 34/5, x + 42/5

Test IV answers:1.x = 3, t21 = -128

2.n = 41st term3.d = 3/2, a = 34.11th term = 89

5.t16 = 55 cm

Prepared by:•Marx Lennin

Cabaltican•Bernadette Aubrey

Cabrera•Precious Fernandez