Sequence a₁, a₂, a₃ … the sequence {a₁ + a₂ + a₃ + … + an is called a series. A series is finite or infinite according as the number of terms added is finite or infinite.
Example: List the first four terms of the sequence {an} = {n²}, starting with n = 1.
{a₁, a₂, a₃, a₄}
= {1², 2², 3², 4²}
= {1, 4, 9, 16}
Progressions: Sequences whose terms follow certain patterns are called progressions.
Arithmetic Progression: An arithmetic progression (AP) is a sequence in which terms increase or decrease regularly by the same constant. A general AP, where a is the first term of AP and d is the common difference of AP.
a, a + d, a + 2d, … a + (n – 1) dGeneral Term: Tn = a + (n – 1) d
If the all terms of an AP are increased, decreased, multiplied and divided by the same non-zero constant, then they remain in AP.
Three consecutive numbers in AP can be taken as a – d, a, a + d.
Example: An arithmetic progression with initial term 4 and common difference 8, then find the 10 terms
Solution: Tn = a + (n – 1) d
a = 4, d = 8
T₁₀ = 4 + (10 – 1) 8
T₁₀ = 76
Sum of AP: \(Sn\,=\,\frac{n}{2}\left[ 2a+\left( n-1 \right)d \right]\,=\,\frac{n}{2}\left[ a+l \right]\)
Arithmetic Mean: If a, A, b are in AP, then A is called by arithmetic mean.
\(A\,=\,\frac{\left( a\,+\,b \right)}{2}\)Geometric Progression: A sequence is said to be a geometric progression (GP), if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio.
a, ar, ar² ….
Example: 1, 2, 4, … 64
This is a finite sequence with a = 1 and r = 2
The geometric progression with unlimited number of terms is called Infinite Sequence. It does not have a last term.1, 3, 9, 27 …
This is an infinite sequence with a = 1 and r = 3
General Term: Tn = arⁿ¯¹
Sum of Terms: \({{S}_{n}}\,=\,\frac{a\left( {{r}^{n}}-1 \right)}{\left( r-1 \right)}\)
Geometric Mean: The geometric mean (GM) of any two positive numbers a and b is given by √ab. The sequence a, G, b is GP.
G = √ab.