![ARITHMETIC SEQUENCE, MEAN & SERIES
a sequence whose
terms have a
common difference
𝒅
𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅
the term that lie
between two
nonconsecutive
terms of an
arithmetic
sequence
the sum of terms
of an arithmetic
sequence
𝑺𝒏 =
𝒏
𝟐
𝒂𝟏 + 𝒂𝒏
𝑺𝒏 =
𝒏
𝟐
[𝟐𝒂𝟏 + 𝒏 − 𝟏 𝒅]
ARITHMETIC SEQUENCE ARITHMETIC
MEAN
ARITHMETIC
SERIES](https://image.slidesharecdn.com/arithmeticsequence-230106003029-69a4ed1e/85/Arithmetic-Sequence-pptx-11-320.jpg)
Arithmetic Sequence.pptx
- 1. At the end of this module, you will learn to: 1. Generate patterns. 2. lllustrate an arithmetic sequence. 3. Determine arithmetic means, nth term of an arithmetic sequence and sum of the terms of a given arithmetic sequence.
- 2. Identify the pattern of each sequence then find the next three terms of each sequence. 1. 2, 5, 8, 11, 14, ___, ___, ___ 2. A, B, C, E, F, G, ___, ___, ___ 3. 74, 69, 64, 59, ___, ___, ___ 4. 1, 4, 16, 64, ____, _____, _____ 5. 2, 3, 5, 8, 12, 17, ___, ___, ___ 17 20 23 I J K 54 49 44 256 1024 4096 23 30 38 *** To identify the next terms, patterns should be determined first. SEQUENCE - a list of objects or numbers arranged in a definite order
- 3. Example: 7, 11, 15, 19, … is a sequence. The pattern is adding 4 to each term. 4th term (𝑎4) 3rd term (𝑎3) 2nd term (𝑎2) 1st term (𝑎1)
- 4. ARITHMETIC SEQUENCE a sequence whose consecutive terms have a common difference 𝒅 (which can be identified by subtracting the term by its previous term)
- 5. 7, 11, 15, 19, … is an arithmetic sequence with a common difference of 4. 7, 11, 15, 19, … 𝑎1 𝑎2 𝑎3 𝑎4 11 − 7 = 4 15 − 11 = 4 19 − 15 = 4
- 6. Determine whether the sequence is an arithmetic sequence or not by finding its common difference. 1. 4, 10, 16, 22, … 2.17, 19, 21, 24, … 3. 5, 2, -1, -4, … The common difference 𝑑 = 6, therefore it is an arithmetic sequence. There is no common difference, therefore it is NOT an arithmetic sequence. The common difference 𝑑 = −3, therefore it is an arithmetic sequence.
- 7. Nth TERM OF AN ARITHMETIC SEQUENCE The nth term of an arithmetic sequence is also called its general term. It is determined by using the formula: 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 where: 𝑎𝑛 = 𝑛𝑡ℎ term 𝑎1 = 1𝑠𝑡 term 𝑛 =number of terms 𝑑 =common difference
- 8. Find the next three terms of each sequence. 1. 7, 11, 15, 19, … 2. -5, -7, -9, -11, … 𝑑 = 𝑎2 − 𝑎1 𝑑 = 11 − 7 𝑑 = 4 7, 11, 15, 19, ___, ___, ___ 23 27 31 𝑑 = 𝑎2 − 𝑎1 𝑑 = −7 − (−5) 𝑑 = −7 + 5 -5, -7, -9, -11, ___, ___, ___ -13 -15 -17 𝑑 = −2
- 9. Solve for the indicated term of each arithmetic sequence. 3. 8th term of the arithmetic sequence 7,11,15,19,… 𝑑 = 𝑎2 − 𝑎1 𝑑 = 11 − 7 𝑑 = 4 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎8 = 7 + 8 − 1 4 𝑎8 = 7 + (7)4 𝑎8 = 7 + 28 𝑎8 = 35
- 10. Solve for the indicated term of each arithmetic sequence. 4. 50th term of the arithmetic sequence -5,-7,-9,-11,… 𝑑 = 𝑎2 − 𝑎1 𝑑 = −7 − (−5) 𝑑 = −7 + 5 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎50 = −5 + 50 − 1 − 2 𝑎50 = −5 + 49 − 2 𝑎50 = −5 + (−98) 𝑎50 = −103 𝑑 = −2
- 11. ARITHMETIC SEQUENCE, MEAN & SERIES a sequence whose terms have a common difference 𝒅 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅 the term that lie between two nonconsecutive terms of an arithmetic sequence the sum of terms of an arithmetic sequence 𝑺𝒏 = 𝒏 𝟐 𝒂𝟏 + 𝒂𝒏 𝑺𝒏 = 𝒏 𝟐 [𝟐𝒂𝟏 + 𝒏 − 𝟏 𝒅] ARITHMETIC SEQUENCE ARITHMETIC MEAN ARITHMETIC SERIES