Sequences and Series

Harmonic sequence formula is the sequence of real numbers determined by taking the reciprocals of the arithmetic progression.

Geometric Progression (GP) | Sequences and Series

GP is a sequence of numbers whose first term is non zero & each of the succeeding terms is equal to the proceeding terms multiplied by a constant . Thus in a GP the ratio of successive terms is constant. This constant factor is called the COMMON RATIO of the series & is obtained by dividing any term by that which immediately proceeds it. Therefore a, ar, ar², ar³, ar⁴, …… is a GP with a as the first term & r as common ratio.

• nth term = ar^{n –1}
• Sum of the Ist n terms i.e. \(S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}, \text { if } r \neq 1\)
• Sum of an infinite GP when |r| < 1 when n → ∞ rⁿ → 0 if |r| < 1 therefore,
  \(S_{\infty}=\frac{a}{1-r}(|r|<1)\)
• If each term of a GP be multiplied or divided by the same non-zero quantity, the resulting sequence is also a GP.
• Any 3 consecutive terms of a GP can be taken as a/r, a, ar; any 4 consecutive terms of a GP can be taken as a/r³, a/r, ar, ar³ & so on.
• If a, b, c are in GP ⇒ b² = ac.

Harmonic Progression (HP) | Sequences and Series

A sequence is said to HP if the reciprocals of its terms are in AP.If the sequence a1, a2, a3, …. , an is an HP then 1/a1, 1/a2, …. , 1/an is an AP & converse. Here we do not have the formula for the sum of the n terms of an HP. For HP whose first term is a & second term is b, the nth term is
\(\mathrm{t}_{\mathrm{n}}=\frac{\mathrm{a} \mathrm{b}}{\mathrm{b}+(\mathrm{n}-1)(\mathrm{a}-\mathrm{b})}\)

If a, b, c are in HP ⇒ \(\mathrm{b}=\frac{2 \mathrm{ac}}{\mathrm{a}+\mathrm{c}} \text { or } \frac{\mathrm{a}}{\mathrm{c}}=\frac{\mathrm{a}-\mathrm{b}}{\mathrm{b}-\mathrm{c}}\)

Sequences and Series | Means

Arithmetic Mean:
If three terms are in AP then the middle term is called the AM between the other two, so if a, b, c are in AP, b is AM of a & c . AM for any n positive number a₁, a₂, … , a_n is ;
\(A = \frac {a_{1}+a{2}+a_{3}+\ldots\ldots+a_{n}}{n}\)
n – Arithmetic Means Between Two Numbers:
If a, b are any two given numbers & a, A₁, A₂, …. , A_n, b are in AP then A₁, A₂, … A_n are the n AM’s between a & b.
\(\mathrm{A}_{1}=\mathrm{a}+\frac{\mathrm{b}-\mathrm{a}}{\mathrm{n}+1}, \mathrm{A}_{2}=\mathrm{a}+\frac{2(\mathrm{b}-\mathrm{a})}{\mathrm{n}+1}, \ldots \ldots, \mathrm{A}_{\mathrm{n}}=\mathrm{a}+\frac{\mathrm{n}(\mathrm{b}-\mathrm{a})}{\mathrm{n}+1} \quad=\mathrm{a}+\mathrm{d}, \quad=\mathrm{a}+2 \mathrm{d}, \ldots \ldots, \mathrm{A}_{\mathrm{n}}=\mathrm{a}+\mathrm{nd}\)
where \(\mathrm{d}=\frac{\mathrm{b}-\mathrm{a}}{\mathrm{n}+1}\)
Note: Sum of n AM’s inserted between a & b is equal to n times the single AM between a & b i.e.
\(\overset{n}{\underset{r=1}{\sum}{A_{r}}}=nA\) where A is the single AM between a & b.

Geometric Means:
If a, b, c are in GP, b is the GM between a & c.
b² = ac, therefore b = \(\sqrt {ac}\) ; a > 0, c > 0.
n-Geometric Means Between a, b:
If a, b are two given numbers & a, G₁, G₂, ….. , G_n, b are in GP. Then
G₁, G2, G₃ , …., G_n are n GMs between a & b
G₁ = a(b/a)^{1 /n+1} ,G₂ = a(b/a)^{2 /n+1} = ar , ……., G_n = a(b/a)^{n /n+1} = ar ,
G₁ = ar ,                G₂ = ar² ,                      ……., G_n = arⁿ ,  where r = (b/a)^{1 /n+1}

Note: The product of n GMs between a & b is equal to the n^th power of the single GM between a & b  i.e.
\(\underset{\mathrm{r}=1}{\pi} \mathrm{G}_{\mathrm{r}}=(\mathrm{G})^{\mathrm{n}}\)
where G is the single GM between a & b.

Harmonic Mean:
If a, b, c are in HP, b is the HM between a & c, then b = 2ac/[a + c].
Theorem:
If A, G, H are respectively AM, GM, HM between a & b both being unequal & positive then,

• G² = AH
• A > G > H (G > 0). Note that A, G, H constitute a GP.

Arithmetic-Geometric Series | Sequences and Series

A series each term of which is formed by multiplying the corresponding term of an AP & GP is called the Arithmetico-Geometric Series. e.g. 1 + 3x + 5x² + 7x³ + …..
Here 1, 3, 5, …. are in AP & 1, x, x², x³ ….. are in GP.
Standard appearance of an Arithmetic-Geometric Series is
Let S_n = a + (a + d) r + (a + 2 d) r² + ….. + [a + (n − 1)d] rⁿ⁻¹
Sum To Infinity:
If |r| < 1 & n → ∞ then \({ \lim_{ _{ n\rightarrow \infty } }{ { r }^{ n } } }=0.\)\(S_{\infty}=\frac{a}{1-r}+\frac{d r}{(1-r)^{2}}\)

Sigma Notations Theorems:

• \(\sum_{r=1}^{n}\left(a_{r} \pm b_{r}\right)=\sum_{r=1}^{n} a_{r} \pm \sum_{r=1}^{n} b_{r}\)
• \(\sum_{r=1}^{n} k a_{r}=k \sum_{r=1}^{n} a_{r}\)
• \(\sum_{r=1}^{n} k=n k\)
  where k is a constant.

Results:

• \(\sum_{r=1}^{n} r=\frac{n(n+1)}{2}\)  (sum of the first n natural nos.)
• \(\sum_{r=1}^{n} r^{2}=\frac{n(n+1)(2 n+1)}{6}\)  (sum of the squares of the first n natural numbers)
• \(\sum_{r=1}^{n} r^{3}=\frac{n^{2}(n+1)^{2}}{4}\left[\sum_{r=1}^{n} r\right]^{2}\)  (sum of the cubes of the first n natural numbers)
• \(\sum_{r=1}^{n} r^{4}=\frac{n}{30}(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)\)

Method Of Difference | Sequences and Series

If T₁, T₂, T₃, …… , T_n are the terms of a sequence then some times the terms T₂ − T₁, T₃ − T₂ , ……. constitute an AP/GP. n^th term of the series is determined & the sum to n terms of the sequence can easily be obtained.
Remember that to find the sum of n terms of a series each term of which is composed of r factors in AP, the first factors of several terms being in the same AP, we “write down the nth term, affix the next factor at the end, divide by the number of factors thus increased and by the common difference and add a constant. Determine the value of the constant by applying the initial conditions”.