Important Concepts and Formulas - Sequence and Series

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




Arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, d to the preceding term. The constant d is called common difference.

An arithmetic progression is given by a, (a + d), (a + 2d), (a + 3d), ...
where a = the first term , d = the common difference

Examples for Arithmetic Progressions

1, 3, 5, 7, ... is an arithmetic progression (AP) with a = 1 and d = 2

7, 13, 19, 25, ... is an arithmetic progression (AP) with a = 7 and d= 6




• n^th term of an arithmetic progression

t_n = a + (n – 1)d

where t_n = n^th term, a= the first term , d= common difference

Example 1 : Find 10^th term in the series 1, 3, 5, 7, ...

a = 1

d = 3 – 1 = 2

10^th term, t₁₀ = a + (n-1)d = 1 + (10 – 1)2 = 1 + 18 = 19

Example 2 : Find 16^th term in the series 7, 13, 19, 25, ...

a = 7

d = 13 – 7 = 6

16^th term, t₁₆ = a + (n-1)d = 7 + (16 – 1)6 = 7 + 90 = 97





• Number of terms of an arithmetic progression

$MF#%n = \dfrac{(l - a)}{d} + 1 $MF#%

where n = number of terms, a= the first term , l = last term, d= common difference

Example : Find the number of terms in the series 8, 12, 16, . . .72

a = 8

l = 72

d = 12 – 8 = 4

$MF#%n = \dfrac{(l - a)}{d} + 1 = \dfrac{(72 - 8)}{4} + 1 = \dfrac{64}{4} + 1 = 16 + 1 = 17$MF#%





• Sum of first n terms in an arithmetic progression

$MF#%\begin{align}& \\\\ & S_n = \dfrac{n}{2} [\ 2a + (n - 1)d\ ] \ = \dfrac{n}{2}[\ a + l\ ]\\\\\\ & \text{where a = the first term,}\quad \text{ d= common difference, }\quad l = t_n = \text{n}^{th}\text{ term = } a + (n-1)d\end{align}$MF#%

Example 1 : Find 4 + 7 + 10 + 13 + 16 + . . . up to 20 terms

a = 4

d = 7 – 4 = 3

Sum of first 20 terms, S₂₀ $MF#% = \dfrac{n}{2} [\ 2a + (n - 1)d\ ] = \dfrac{20}{2} [\ (2 \times 4) + (20 - 1)3\ ] $MF#%

$MF#% = 10 [8 + (19 \times 3) ] = 10 [8 + 57 ] = 650 $MF#%





Example 2 : Find 6 + 9 + 12 + . . . + 30

a = 6

l = 30

d = 9 – 6 = 3

$MF#%n = \dfrac{(l - a)}{d} + 1 = \dfrac{(30 - 6)}{3} + 1 = \dfrac{24}{3} + 1 = 8 + 1 = 9$MF#%

$MF#%\text{Sum, S = }\dfrac{n}{2}[\ a + l\ ] = \dfrac{9}{2}[\ 6 + 30\ ] = \dfrac{9}{2} \times 36 = 9 \times 18 = 162$MF#%




• Arithmetic Mean

If a, b, c are in AP, b is the Arithmetic Mean (A.M.) between a and c.

In this case, $MF#% b = \dfrac{1}{2}(a + c)$MF#%
The Arithmetic Mean (A.M.) between two numbers a and b = $MF#%\dfrac{1}{2}(a + b)$MF#%
If a, a₁, a₂ ... a_n, b are in AP we can say that a₁, a₂ ... a_n are the n Arithmetic Means between a and b.
• If a, b, c are in AP, 2b = a + c
• To solve most of the problems related to A.P., the terms can be conveiently taken as
  
  3 terms : (a – d), a, (a +d)
  
  4 terms : (a – 3d), (a – d), (a + d), (a +3d)
  
  5 terms : (a – 2d), (a – d), a, (a + d), (a +2d)
• T_n = S_n - S_n-1
• If each term of an A.P. is increased, decreased , multiplied or divided by the same non-zero constant, the resulting sequence also will be in A.P.
• In an A.P., sum of terms equidistant from beginning and end will be constant



• Harmonic Progression (H.P.)  (or Harmonic Sequence)   




$MF#%\text{Non-zero numbers }a_1,\ a_2,\ a_3,\ ... \ a_n\text{ are in H.P. if }\dfrac{1}{a_1},\ \dfrac{1}{a_2},\ \dfrac{1}{a_3},\ \text{ ... }\dfrac{1}{a_n}\text{ are in A.P.}$MF#%

Examples for Harmonic Progressions

$MF#%\dfrac{1}{2}, \dfrac{1}{6}, \dfrac{1}{10}, \cdots $MF#% is a harmonic progression (H.P.)

Three non-zero numbers a, b, c will be in HP, if  $MF#%\dfrac{1}{a},\ \dfrac{1}{b},\ \dfrac{1}{c}$MF#% are in A.P.

If a, (a+d), (a+2d), . . . are in A.P., n^thterm of the A.P. = a + (n - 1)d

Hence, if $MF#%\dfrac{1}{a}, \dfrac{1}{a+d}, \dfrac{1}{a+2d}, \cdots$MF#% are in H.P., n^thterm of the H.P. = $MF#%\dfrac{1}{a + (n - 1)d}$MF#%

If a, b, c are in HP, b is the Harmonic Mean(H.M.) between a and c

In this case, $MF#% b = \dfrac{2ac}{a + c}$MF#%

The Harmonic Mean (H.M.) between two numbers a and b = $MF#%\dfrac{2ab}{a + b}$MF#%

If a, a₁, a₂ ... a_n, b are in H.P. we can say that a₁, a₂ ... a_n are the n Harmonic Means between a and b.

If a, b, c are in HP, $MF#%\dfrac{2}{b} = \dfrac{1}{a} + \dfrac{1}{c}$MF#%



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




A sequence of non-zero numbers is a Geometric Progression (G.P.) if the ratio of any term and its preceding term is always constant.

A Geometric Progression (G.P.) is given by a, ar, ar², ar³, ...
where a = the first term , r = the common ratio

Examples for Geometric Progressions

1, 3, 9, 27, ... is a geometric progression (G.P.) with a = 1 and r = 3

2, 4, 8, 16, ... is a geometric progression (G.P.) with a = 2 and r = 2




• n^th term of a geometric progression (G.P.)

$MF#%t_n = ar^{n-1}$MF#%

where t_n = n^th term, a= the first term , r = common ratio, n = number of terms

Example 1 : Find the 10^th term in the series 2, 4, 8, 16, ...

a = 2,    r = $MF#%\dfrac{4}{2}$MF#% = 2,   n = 10

10^th term, t₁₀ = $MF#%ar^{n-1} = 2 \times 2^{10-1} = 2 \times 2^{9} = 2 \times 512 = 1024$MF#%




Example 2 : Find 5^th term in the series 5, 15, 45, ...

a = 5,   r = $MF#%\dfrac{15}{5}$MF#% = 3,   n = 5

5^th term, t₅ = $MF#%ar^{n-1} = 5 \times 3^{5-1}$MF#% = 5 × 3⁴ = 5 × 81 = 405





• Sum of first n terms in a geometric progression (G.P.)

$MF#%S_n = \begin{cases} \dfrac{a(r^n - 1)}{r - 1}\ & \text{(if r > 1)} \\ \\ \\ \\ \\ \\ \\ \\ \dfrac{a(1 - r^n)}{1 - r}\ & \text{(if r < 1)} \\ \end{cases}$MF#%

where a= the first term , r = common ratio, n = number of terms




Example 1 : Find 4 + 12 + 36 + ... up to 6 terms

a = 4,   r = $MF#%\dfrac{12}{4}$MF#% = 3,   n = 6

Here r > 1. Hence, $MF#%S_6 = \dfrac{a(r^n - 1)}{r - 1} = \dfrac{4(3^6 - 1)}{3 - 1} = \dfrac{4(729 - 1)}{2} = \dfrac{4 \times 728}{2} = 2 \times 728 = 1456$MF#%




Example 2 : Find $MF#%1 + \dfrac{1}{2} + \dfrac{1}{4} +$MF#% ... up to 5 terms

a = 1,   r = $MF#%\dfrac{\left(\dfrac{1}{2}\right)}{1} = \dfrac{1}{2},$MF#%   n = 5

Here r < 1. Hence, $MF#%S_6 = \dfrac{a(1 - r^n)}{1 - r} = \dfrac{1\left[1 - \left(\dfrac{1}{2}\right)^5 \right]}{\left(1 - \dfrac{1}{2}\right)} = \dfrac{\left(1 - \dfrac{1}{32} \right)}{\left(\dfrac{1}{2}\right)} = \dfrac{\left(\dfrac{31}{32}\right) }{\left(\dfrac{1}{2}\right)} = \dfrac{31}{16} = 1\dfrac{15}{16}$MF#%




• Sum of an infinite geometric progression (G.P.)

$MF#%S_\infty = \dfrac{a}{1 - r}\ \text{ (if    0 < r < 1)}$MF#%

where a= the first term , r = common ratio



Example$MF#%\text{ : Find }1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} +\text{ . . . }\infty$MF#%

a = 1,       r = $MF#%\dfrac{\left(\dfrac{1}{2}\right)}{1} = \dfrac{1}{2}$MF#%

Here 0 < r < 1. Hence, $MF#%S_\infty = \dfrac{a}{1 - r} = \dfrac{1}{\left(1 - \dfrac{1}{2}\right)} = \dfrac{1}{\left(\dfrac{1}{2}\right)} = 2$MF#%

• Geometric Mean

If three non-zero numbers a, b, c are in G.P., b is the Geometric Mean (G.M.) between a and c.

In this case, $MF#% b = \sqrt{ac}$MF#%
The Geometric Mean (G.M.) between two numbers a and b = $MF#%\sqrt{ab}$MF#%

(Note that if a and b are of opposite sign, their G.M. is not defined.)
• If a, b, c are in G.P., b² = ac
• If a, b, c are in G.P., $MF#%\dfrac{a-b}{b-c} = \dfrac{a}{b}$MF#%
• In a G.P., product of terms equidistant from beginning and end will be constant.
• To solve most of the problems related to G.P., the terms of the G.P. can be conveiently taken as
  
  3 terms : $MF#%\dfrac{a}{r}$MF#%, a, ar
  
  5 terms : $MF#%\dfrac{a}{r^2}$MF#%, $MF#%\dfrac{a}{r}$MF#%, a, ar, ar²



• Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers   



If GM, AM and HM are the Geometic Mean, Arithmetic Mean and Harmonic Mean of two positive numbers respectively, then

GM² = AM × HM

• Some Interesting Properties to Note   



Three numbers a, b and c are in AP if $MF#%b = \dfrac{a + c}{2}$MF#%

Three non-zero numbers a, b and c are in HP if $MF#%b = \dfrac{2ac}{a + c}$MF#%

Three non-zero numbers a, b and c are in HP if $MF#%\dfrac{a - b}{b - c} = \dfrac{a}{c}$MF#%
Let A, G and H be the A.M., G.M. and H.M. between two distinct positive numbers. Then


• A > G > H


• A, G and H are in GP
If a series is both an A.P. and G.P., all terms of the series will be equal. In other words, it will be a constant sequence.



• Power Series : Important formulas   






• $MF#%1 + 1 + 1 + \cdots\text{ n terms}= \sum{1} = n%$MF#%




• $MF#%1 + 2 + 3 + \cdots + n = \sum{n} = \dfrac{n(n+1)}{2}%$MF#%




• $MF#%1^2 + 2^2 + 3^2 + \cdots + n^2 = \sum{n^2} = \dfrac{n(n+1)(2n + 1)}{6}%$MF#%




• $MF#%1^3 + 2^3 + 3^3 + \cdots + n^3 = \sum{n^3} = \dfrac{n^2(n+1)^2}{4} = \left[\dfrac{n(n+1)}{2}\right]^2 %$MF#%

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