Introduction:
Sequence: A sequence is a function whose domain is the set N of natural numbers.
Real Sequence: A sequence whose range is a subset of R is called a real sequence.
In other words, a real sequence is a function with domain N and the range a subset of the set R of real numbers.
Series: If a₁, a₂, a₃, a₄ … an is a sequence, then the expression a₁ + a₂ + a₃ + a₄ + a₅ + … + an + … is a series.
Progressions: It is not necessary that the terms of a sequence always follow a certain pattern or they are described by some explicit formula for the nth term. Those sequences whose terms follow certain pattern are called progressions.
i) Every Progression is a series but every Series need not be a Progression.
ii) They are 4 types of Progressions (mainly 3)
1) Arithmetic Progression
2) Geometric Progression
3) Harmonic Progression
4) Arithmetic-Geometric Progression
Arithmetic Progression (A.P): A sequence is called an arithmetic progression if the difference of a term and the previous term is always same i.e. an₊₁ – an = constant (=d) for all n ϵ N. The constant difference, generally denoted by d is called the common difference.
Illustration 1: 1, 4, 7, 10 … is an A.P. whose first term is 1 and the common difference is 4 – 1 = 3.
Illustration 2: 11, 7, 3, -1 … is an A.P. whose first term is 11 and the common difference 7 – 11 = -4.
Properties of an Arithmetic Progression:
Property I: If a is the first term and d the common difference of an A.P., then its nth terms an is given by an = a + (n – 1) d.
Property II: A sequence is an A.P if its nth term is of the form case is An+ B i.e. a linear expression in n. The common difference is such a case is A i.e. the coefficient of n.
Property III: If a constant is added to or subtracted from each term of an A.P., then the resulting sequence is also an A.P. with the same common difference.
Property IV: If each term of a given A.P. is multiplied or divided by a non-zero constant k, then the resulting sequence is also an A.P. with common difference kd or d/k , where d is the common difference of the given AP.
Property V: In a finite A.P. the sum of the terms equidistant from the beginning and end is always same and is equal to the sum of first and last term i.e. ak + an₋(k₋₁) = a₁ + an.
for all k = 1 ,2 ,3 ,…, n-1.
Property VI: Three numbers a, b, c are in A.P. if 2b = a + c.
Property VII: If the terms of an A.P. are chosen at regular intervals, then the form of an A.P.
Property VIII: If an, an+1 and an+2 three consecutive terms of an A.P., then, 2an₊₁ = an + an₊₂.
Selection of Terms in an A.P: The following ways of selecting terms are generally very convenient.
|
Number of Terms |
Terms | Common Difference |
|
3 |
a – d, a, a + d | d |
| 4 | a – 3d, a – d, a + d, a + 3d |
2d |
| 5 |
a – 2d, a – d, a, a + d, a + 2d |
d |
| 6 | a – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d |
2d |
Some Useful Results:
\(\sum\limits_{r=1}^{n}{r}=1+2+…+n=\frac{n(n+1)}{2}\).
\(\sum\limits_{r=1}^{n}{{{r}^{2}}}={{1}^{2}}+{{2}^{2}}+…+{{n}^{2}}=\frac{n(n+1)\left( 2n+1 \right)}{6}\).
\(\sum\limits_{r=1}^{n}{{{r}^{3}}}={{1}^{3}}+{{2}^{3}}+…+{{n}^{3}}={{\left\{ \frac{n(n+1)}{6} \right\}}^{2}}\).
\(\sum\limits_{r=1}^{n}{{{r}^{4}}}={{1}^{4}}+{{2}^{4}}+…+{{n}^{4}}=\frac{n(n+1)(6{{n}^{3}}+9{{n}^{2}}+n-1)}{30}\).
An Important Property: A sequence is an A.P. if and only if the sum of its n terms is of the form An² + Bn. Where A, B are constants. In such a case, the common difference of the A.P is 2A.
Remark: It follows from this property that a sequence is an A.P. if the sum of its n terms is of the form An² + Bn. i.e., a quadratic expression in n and in such a case the common difference is twice the coefficient of n2. For example, if Sn = 3n² + 2n. We can say that it is sum of n terms of an A.P with common difference 6.
Insertion of Arithmetic Means: If between two given quantities a and b we have to insert n quantities A₁, A₂ … An. such that A₁, A₂ … An, b form an A.P. Then we say that A₁, A₂ … An are arithmetic means between a and b.
Example: Since 15, 11, 7, 3, -1, -5 are in A.P, it follows that 11, 7, 3, -1 are four arithmetic means between 15 and -5.
Insertion of n Arithmetic Means between a and b: Let A₁, A₂ … An be n arithmetic means between a and b. Then a, A₁, A₂ … An, b is an A.P. Let d be the common difference of this A.P. Clearly, it contains (n+2) terms.
∴ b = (n + 2)th term
b = a + (n + 1)d
Now,
\({{A}_{1}}=\left( a+\frac{b-a}{n+1} \right)\).
A₂ = a + 2d ⇒ \({{A}_{2}}=\left( a+\frac{2\left( b-a \right)}{n+1} \right)…\).
An = a + nd ⇒ \({{A}_{n}}=\left( a+\frac{n\left( b-a \right)}{n+1} \right).\).
These are the required arithmetic means between a and b.
Insertion of a Single Arithmetic Mean between a and b: Let A be the arithmetic mean of a and b. Then a, A, b are in A.P.
A – a = b – A
2A = a + b
\(A=\frac{a+b}{2}\).