Chapter 6: Sequence and Series

• Sequence

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  • Arrangement of numbers subject to some definite rule.
  • A function whose domain is a subset of Natural Numbers and range is Real or Complex Numbers ($f:N \to \mathbb{R}/C$).
  • Notation: $\{a_n\}$.
  • Each number in sequence is called a Term.
  • The $n^{th}$ term or general term of a sequence is denoted by $a_n$.
  • $a_1, a_2, a_3, a_4$ be the first, second, third and, fourth terms of a sequence respectively.
  • Real Sequence: If all members of a sequence are real numbers.
  • Finite Sequence: If the domain of a sequence is a finite set.
  • Infinite Sequence: If the domain of a sequence is an infinite set (has no last term).

• Arithmetic Sequence or Arithmetic Progression (A.P.)

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  • A sequence in which every term after the first is obtained from the preceding term by adding a fixed number in it.
  • Common Difference (d): The difference of two consecutive terms ($a_n - a_{n-1} = d$, where $n>1$). $d$ can be any number.
  • General form of an A.P.: $a_1, a_1+d, a_1+2d, \dots, a_1+(n-1)d$.
  • General Term or $n^{th}$ Term: $a_n = a_1 + (n-1)d$.
    • $a_1$: First term.
    • $n$: Number of terms (always a positive integer).
    • $d$: Common Difference.
    • $a_n$: $n^{th}$ term or general term.

  • Arithmetic Mean (A.M.)

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    • If three terms are in A.P., then the middle term is known as Arithmetic Mean (A.M.) of the other two terms.
    • If $a, A, b$ are in A.P., then $A = \frac{a+b}{2}$.
    • If $\alpha, \beta, \gamma$ are in A.P., then their A.M. is $\beta$.

  • n Arithmetic Means Between Two Numbers

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    • If $a, A_1, A_2, A_3, \dots, A_n, b$ are in A.P., then the numbers $A_1, A_2, A_3, \dots, A_n$ are called n Arithmetic Means between 'a' and 'b'.
    • $n^{th}$ arithmetic mean between $a$ and $b$: $A_m = a + \frac{m(b-a)}{n+1}$.
    • The sum of n A.M.'s between any two numbers $a$ and $b$ is equal to n-times their A.M.: $A_1 + A_2 + A_3 + \dots + A_n = n\left(\frac{a+b}{2}\right)$.

• Arithmetic Series

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  • The sum of an indicated number of terms of a sequence is called a Series.
  • If $a_1, a_1+d, a_1+2d, a_1+3d, \dots, a_1+(n-1)d$ are in A.P., then the series $a_1+(a_1+d)+(a_1+2d)+\dots+[a_1+(n-1)d]$ is called an Arithmetic Series.

  • Sum of First 'n' Terms of an Arithmetic Series

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    • If $a_1$ is first term, and $a_n=l$ is general term: $S_n = \frac{n}{2}(a_1+a_n)$ or $S_n = \frac{n}{2}(a_1+l)$.
    • The sum of first n terms of arithmetic series can be calculated by using the formula: $S_n = \frac{n}{2}\{2a_1+(n-1)d\}$.
    • If $d=0$ then the above Series becomes $a_1+a_1+a_1+\dots+n$ terms and $S_n = na_1$.

  • Summary of Formulae in A.P.

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    ARITHMETIC PROGRESSION	ARITHMETIC MEAN	ARITHMETIC SERIES
    1. $a_n = a_1 + (n-1)d$	1. $A = \frac{a+b}{2}$	1. $S_n = \frac{n}{2}[2a_1+(n-1)d]$
    2. $d = a_n - a_{n-1}$, $n>1$	2. $A_m = a + \frac{m(b-a)}{n+1}$	2. $S_n = \frac{n}{2}(a_1+a_n)$
    3. $n = \frac{a_n-a_1}{d}+1$	3. $A_1+A_2+A_3+\dots+A_n = n(\frac{a+b}{2})$	3. $S_n = \frac{n}{2}(a_1+l)$

• Geometric Sequence or Geometric Progression (G.P.)

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  • A sequence in which every term after the first is obtained from the preceding term by multiplying it with a non-zero constant number.
  • Common Ratio (r): The constant ratio of any two consecutive terms ($r = \frac{a_n}{a_{n-1}}$, where $n>1$).
  • No term of a G.P. can be zero.
  • General form of a G.P.: $a_1, a_1r, a_1r^2, a_1r^3, \dots, a_1r^{n-1}$.
  • General Term or $n^{th}$ Term of G.P.: $a_n = a_1r^{n-1}$ ($a_n \ne 0$).
    • $a_1$: First term.
    • $r$: Common Ratio.
    • $a_n$: General or $n^{th}$ term.

  • Geometric Mean (G.M.)

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    • If three terms are in G.P., then the middle term is known as Geometric Mean (G.M.) of the other two terms.
    • If $a, G, b$ are in G.P., then $G = \pm\sqrt{ab}$.

  • n Geometric Means Between Two Numbers

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    • If $a, G_1, G_2, G_3, \dots, G_n, b$ are in G.P., then the numbers $G_1, G_2, G_3, \dots, G_n$ are called n geometric means between 'a' and 'b'.
    • $n^{th}$ G.M. between $a$ and $b$: $G_n = a\left(\frac{b}{a}\right)^{\frac{n}{n+1}}$.
    • The product of n G.M.'s between $a$ and $b$ is $n^{th}$ power of single G.M between $a$ and $b$: $G_1 \cdot G_2 \cdot G_3 \dots G_n = (\sqrt{ab})^n$.

• Geometric Series

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  • If $a_1, a_1r, a_1r^2, a_1r^3, \dots, a_1r^{n-1}$ are in G.P., then the series $a_1+a_1r+a_1r^2+a_1r^3+\dots+a_1r^{n-1}$ is called Geometric series.

  • Sum of First 'n' Terms of a Geometric Series (Finite Series)

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    • If $|r|<1$: $S_n = \frac{a_1(1-r^n)}{1-r}$ or $S_n = \frac{a_1-lr}{1-r}$.
    • If $|r|>1$: $S_n = \frac{a_1(r^n-1)}{r-1}$ or $S_n = \frac{lr-a_1}{r-1}$.

  • The Infinite Geometric Series

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    • The series $a_1+a_1r+a_1r^2+a_1r^3+\dots+a_1r^{n-1}+\dots\infty$ is called Infinite Geometric Series.
    • Sum of an Infinite Geometric Series: $S_{\infty} = \frac{a_1}{1-r}$, provided $|r|<1$.
    • Convergent Series: An infinite series is said to be convergent if $n^{th}$ term of series tends to a definite number L as $n\rightarrow\infty$ (i.e., $lim_{n\rightarrow\infty}a_n=L$).
    • Divergent Series: An infinite series is said to be divergent if $n^{th}$ term of series tends to $\pm\infty$ as $n\rightarrow\infty$ (i.e., $lim_{n\rightarrow\infty}a_n=\pm\infty$).
    • If $r=1$, then the series becomes $a_1+a_1+a_1+a_1+\dots$. In this case $S_n$ does not tend to a limit when $n\rightarrow\infty$ and the series does not converge.
    • If $r=-1$, then the series becomes $a_1-a_1+a_1-a_1+a_1-a_1+\dots$.
      • $S_n = a_1$ if $n$ is positive odd integer.
      • $S_n = 0$ if $n$ is positive even integer.
      Thus $S_n$ does not tend to a definite number when $n\rightarrow\infty$. In such a case we say that series is Oscillatory.

  • Summary of Formulae in G.P.

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    GEOMETRIC PROGRESSION	GEOMETRIC MEAN	GEOMETRIC SERIES
    1. $a_n = a_1r^{n-1}$	1. $G = \pm\sqrt{ab}$	For Finite Series:
    2. $r = \frac{a_n}{a_{n-1}}$, $r>1$	2. $G_n = a\left(\frac{b}{a}\right)^{\frac{n}{n+1}}$	1. $S_n = \frac{a_1(1-r^n)}{1-r}$ ($|r|<1$)
    	3. $G_1 \cdot G_2 \cdot G_3 \dots G_n = (\sqrt{ab})^n$	2. $S_n = \frac{a_1(r^n-1)}{r-1}$ ($|r|>1$)
    		For Infinite Series:
    		3. $S_{\infty} = \frac{a_1}{1-r}$ ($|r|<1$)

• Harmonic Progression (H.P.)

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  • A sequence of numbers is called a Harmonic Sequence or Harmonic Progression if the reciprocals of its terms are in Arithmetic Progression.
  • General term of H.P.: $a_n = \frac{1}{a_1+(n-1)d}$.
  • Every H.P. has a corresponding A.P.
  • No term of H.P. can be zero.
  • There is no general method to find the sum of H.P.

  • Harmonic Mean (H.M.)

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    • If three terms are in H.P., then the middle term is called Harmonic Mean (H.M.) of the other two terms.
    • If $a, H, b$ are in H.P., then $H = \frac{2ab}{a+b}$.

  • n Harmonic Means Between Two Numbers

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    • If $a, H_1, H_2, H_3, \dots, H_n, b$ are in H.P., then the numbers $H_1, H_2, H_3, \dots, H_n$ are called n Harmonic Means between $a$ and $b$.
    • $n^{th}$ harmonic mean between $a$ and $b$: $H.M. = \frac{(n+1)ab}{na+b}$.

• Relations between A.M., G.M., and H.M.

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  • $G^2 = A \cdot H$. (A, G, H are in G.P.).
  • If $a$ and $b$ are two distinct positive real numbers and $G>0$: $A > G > H$.
  • If $a$ and $b$ are two distinct negative real numbers and $G<0$: $A < G < H$.
  • If $a=b$: $A=G=H$.

• Sigma Notation or Summation Notation

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  • Sum of $1^{st}$ $n$ natural numbers: $1+2+3+\dots+n = \sum_{k=1}^{n}k = \frac{n(n+1)}{2}$.
  • Sum of squares of $1^{st}$ $n$ natural numbers: $1^2+2^2+3^2+\dots+n^2 = \sum_{k=1}^{n}k^2 = \frac{n(n+1)(2n+1)}{6}$.
  • Sum of cubes of $1^{st}$ $n$ natural numbers: $1^3+2^3+3^3+\dots+n^3 = \sum_{k=1}^{n}k^3 = \left(\frac{n(n+1)}{2}\right)^2 = \frac{n^2(n+1)^2}{4} = \left(\sum_{k=1}^{n}k\right)^2$.
  • $\sum_{k=1}^{m}[k^m-(k-1)^m]=n^m$. (Note: The PDF has 'm' in the summation limit and exponent, which seems like a typo and should likely be 'n' for consistency with the result $n^m$. I've kept it as per the PDF for accuracy to source, but flagging this for your awareness).
  • $1+1+1+\dots+n~\text{terms} = \sum_{k=1}^{n}1 = n$.

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