Question

Provide some important concepts of *Sequence and Series*, class 11th. Please dont spam, or I will report.

Answer 1

Sequence and Series

• A sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it of the type {1, 2, 3, ..., k}.
• A sequence containing finite number of terms is called a finite sequence otherwise it is called infinite series.
• Let a₁, a₂, a₃, ..., aₙ, be a given sequence.

→ Then the expression  a₁ + a₂ + a₃ + ... + aₙ + ... is called the series associated with the given sequence.

• A sequence a₁, a₂, a₃, ..., aₙ, is called arithmetic sequence if aₙ₊₁ = aₙ + d, n ∈ N.

→ Where a₁ = first term, d = common difference

• If a fixed number is added to (or subtracted from) each term of a given AP, then the resulting sequence is also an AP and it has the same common difference as that of the given AP.
• If each term of an AP is multiplied by a fixed constant (or divided by a non-zero fixed constant), then the resulting sequence is also an AP.
• The nth term (general term) of the AP is denoted by aₙ.

→ aₙ = a + (n - 1)d

• If a, a + d, a + 2d, ..., a + (n - 1)d is a given AP, then l = a + (n - 1)d.

$\bullet\ \sf{S_n=\dfrac{n}{2}[2a+(n-1)d]=\dfrac{n}{2}(a+l)}$

→ Sₙ = Sum to n terms of AP, n = number of terms in AP, d = common difference, a = the first term, l = the last term

• If a number A is inserted between 2 numbers a & b so that a, A₁, A₂, A₃, ..., Aₙ, b is an AP, then :

$\sf{A_n=a+nd=a+\dfrac{n(b-a)}{n+1}}$

• A sequence a₁, a₂, a₃, ..., aₙ, is called geometric progression, if each term is non-zero and

$\sf{ \dfrac{a_{k+1}}{a_k}=r\ (constant) , for\ k \ge1 }$

• a, ar, ar², ar³, ... are in GP, where a = first term, r = common ratio
• If each term of a GP is multiplied (or divided) by a fixed non-zero constant, then the resulting sequence is also a GP.
• The nth term of a GP is given by aₙ = arⁿ⁻¹.
• Let the first term of a GP be a and the common ratio be r. Then,

$\sf{S_n=\dfrac{a(1-r^n)}{1-r}\:\:\:\: r<1}$

$\sf{S_n=\dfrac{a(r^n-1)}{1-r} \:\:\:\: r>1}$

• The geometric mean of 2 positive numbers a and b is √ab.
• Let G₁, G₂, G₃, ..., Gₙ be n numbers between positive numbers a and b such that a, G₁, G₂, G₃, ..., Gₙ, b is a GP. Then,

$\sf{G_n=ar^n=a \bigg(\dfrac{b}{a} \bigg)^{\frac{n}{n+1}}}$

• Let A and G be AM and GM of 2 given positive real numbers a and b, respectively. Then, A ≥ G.
• Sum of first n natural numbers is given as :

$\sf{S_n=\dfrac{n(n+1)}{2} }$

• Sum of squares of first n natural numbers is given as :

$\sf{S_n=\dfrac{n(n+1)(2n+1)}{6} }$

• Sum of cubes of the first n natural numbers is given as :

$\sf{S_n=\dfrac{n^2(n+1)^2}{4} }$

• A sequence a₁, a₂, a₃, ..., aₙ, is called harmonic progression if the sequence :

$\displaystyle \sf{\frac{1}{a_1} ,\frac{1}{a_2} ,\frac{1}{a_3},\frac{1}{a_4},...,\frac{1}{a_n}\ is\ in\ AP. }$

• The nth term of a HP is given by :

$\sf{a_n=\dfrac{1}{a+(n-1)d} ,\ where\ a=\dfrac{1}{a_1}. }$