[pic 7]S E Q U E N C E S - A sequence is an ordered collection of objects in which repetitions are allowed.
- It is a function whose domain is the finite set {1,2,3,…,n} where n is the last term, or the infinite set {1,2,3,…} where the infiniteness is denoted by a [… ] called ellipsis.
Examples: a) 4, 8, 12, 16, 20 → Finite Set b) 7, 14, 21, … → Infinite Set
How to find the terms given the general equation to form a sequence: - Substitute each value of n (number of term) in the general equation of the nth term.
Example: a) Given an= 5n-2 as the general equation of the nth term, find its first 5 terms. a1= 5(1)-2 = 3 a2= 5(2)-2 = 8 a3= 5(3)-2 = 13 a4= 5(4)-2 = 18 a5= 5(5)-2 = 23 ∴ {3, 8, 13, 18, 23} is the first 5 terms of the general equation an= 5n-2 which forms a sequence.
TYPES OF SEQUENCES:
- is a sequence where every term after the first term is obtained by adding a constant called the common difference (d).
- a young math student named Carl Gauss created a formula to help solve for the sum of arithmetic sequences. He was born in 1777 in a German Empire and at just ten years old he created this formula.
Example: 6 ∨ 11 ∨ 16 ∨ 21 ∨ 26 → d= 5[pic 8] +5 +5 +5 +5 - In general, the nth term of an Arithmetic sequence is an = a1 + (n-1) d wherein;
■ an = last term ■ n = number of terms ■ a1 = first term ■ d = common difference
Example: a) What is the 21st term of the sequence 5, 9, 13, 17 , … ? an= a1+(n-1)d a21= 5+(21-1)4 a21= 5+(20)4[pic 9] a21= 5+80 a21= 85 |
- ARITHMETIC MEANS- are the terms between any two non-collinear terms of an arithmetic sequence.
- In getting the sum of the terms in an arithmetic sequence these two formulas should be used (depending on the given):
[pic 10][pic 11][pic 12][pic 13]
Example: Give the sum of all the even integers from 1 to 50. an= a1+(n-1)d[pic 14] 50=2+(n-1)2 48=2n-2 50=2n 25=n |
- GEOMETRIC SEQUENCE[pic 15]
- is a sequence where each term after the first term is obtained by multiplying the preceding term by a nonzero constant called the common ratio (r).
- are popularly found in Book IX of Elements by Euclid in 300 B.C. Euclid of Alexandria, a Greek mathematician also considered the "Father of Geometry" was the main contributor of this theory.
Example: 3 ∨ 9 ∨ 27 ∨ 81 →r= 3[pic 16] x3 x3 x3 - Generally, the nth term of a geometric sequence is an= a1rn-1 wherein;
■ an = last term ■ n = number of terms ■ a1 = first term ■ r = common ratio
Example: What is the 6th term of the geometric sequence 5, 25, 125, … ? - a6= ?[pic 17][pic 18]
- a1= 5[pic 19]
- r= 5
- n= 6
- GEOMETRIC MEANS- are the terms between any two given terms of a geometric sequence.
- In getting the sum of the FINITE terms in a geometric sequence, the following conditions must be followed:
- If r = 1 ; iii. If r = -1 : n = odd ; [pic 20][pic 21]
[pic 22] - If r ≠ 1 ; iv. If r = -1 : n = even ; [pic 23]
[pic 24]
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Examples: a) Find the sum of the first 5 terms of the geometric sequence 2, 8, 24, … . c) Find the sum of the first 7 terms of 3, -3, 3, …
d) Find the sum of the first 18 terms of -9, 9, -9, … Sn= a1(1-rn)[pic 25] 1-r[pic 26][pic 27][pic 28] Sn= 2(1-45) 1-4[pic 29] Sn= -2046 -3[pic 30] |
b) Find the sum of 4, 4, 4,…, a7 [pic 31][pic 32][pic 33][pic 34][pic 35][pic 36][pic 37][pic 38][pic 39][pic 40]
- Geometric Sum to infinity only has two conditions:
- Whenever r ≤ -1 or r ≥ 1 the sum of the terms of an infinite geometric sequence does not exist.
- Whenever -1, < r < 1 the formula to be used is [pic 41][pic 42]
Examples: a) Find the sum to infinity of the geometric sequence 3, 6, 12, … (if it exists) - r = 2[pic 43][pic 44][pic 45]
b) Find the sum to infinity of the geometric sequence 2, 2, 2, … (if it exists)
- is a sequence such that the reciprocals of the terms form an arithmetic sequence.
- The study of harmonic sequences dates to at least the 6th century bce, when the Greek philosopher and mathematician Pythagoras and his followers sought to explain through numbers the nature of the universe.
Examples: a) 2, 4, 6, 8, 10 → Arithmetic Sequence[pic 47] 1/2, ¼, 1/6, 1/8, 1/10 → Harmonic Sequence
- a sequence where the first two terms are either both 0 and 1, or 1; and each term therefore is obtained by adding the two preceding terms.
- was invented by the Italian Leonardo Pisano Bigollo (1180-1250), who is known in mathematical history by several names: Leonardo of Pisa (Pisano means "from Pisa") and Fibonacci (which means "son of Bonacci").
- is probably the most famous sequence in the Mathematics world.
Example: 0, ∨ 1, ∨ 1, ∨ 2, ∨ 3, ∨ 5, ∨ 8, ∨ 13, …[pic 48] + + + + + + +
[pic 49] Clare Millen A. Manuel 10-Venus |
