Sequence
A sequence is an ordered list of numbers. Each number in the sequence is called a term.
Example: {2,4,6,8,10}
Finite Sequence
A finite sequence is a sequence that has a limited number of terms. For example, the sequence 2, 4, 6, 8 is finite because it has 4 terms.
Infinite Sequence
An infinite sequence is a sequence that continues indefinitely without terminating. For example, the sequence 1, 2, 3, 4, 5, ... goes on forever.
Arithmetic Sequence
An arithmetic sequence is a sequence in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. For example, the sequence 3, 6, 9, 12, ... is arithmetic with a common difference of 3.
Geometric Sequence
A geometric sequence is a sequence in which the ratio between any two consecutive terms is constant. This constant ratio is called the common ratio. For example, the sequence 2, 4, 8, 16, ... is geometric with a common ratio of 2.
Finite Arithmetic Sequence
A finite arithmetic sequence has a limited number of terms and each term increases (or decreases) by a constant amount. For example, 5, 10, 15, 20 is a finite arithmetic sequence with 4 terms and a common difference of 5.
Finite Geometric Sequence
A finite geometric sequence has a limited number of terms and each term is multiplied by a constant factor. For example, 3, 9, 27 is a finite geometric sequence with 3 terms and a common ratio of 3.
Infinite Arithmetic Sequence
An infinite arithmetic sequence goes on indefinitely with a constant difference between terms. For example, the sequence 4, 7, 10, 13, ... is infinite with a common difference of 3.
Infinite Geometric Sequence
An infinite geometric sequence goes on indefinitely with a constant ratio between terms. For example, the sequence 1, 0.5, 0.25, 0.125, ... is infinite with a common ratio of 0.5.
Formula: Arithmetic Sequence
In the case of a parallel sequence each subsequent element is obtained by adding a fixed number (called the difference d) to the preceding element.
Find the nth term;
an = a + (n - 1)d
where,
an = n-th term
a = First Term
d = Common difference
n = number of terms
Example (Finite Sequence): {3,6,9,12,15}
a = 3 and d = 3
a5 = 3 + (5 - 1)⋅3 = 15
Example (Infinite Sequence): {1,4,7,10,13,…}
a = 1 and d = 3
an = 1 + (n - 1)⋅3 = 3n - 2
Formula: Sum of Finite Arithmetic Sequence
Sn = n2{2a + (n - 1)d}
Example: {3,6,9,12,15}
an = n-th term
a = First Term
d = Common difference
n = number of terms
Example (Finite Sequence): {3,6,9,12,15}
a = 3 and d = 3
a5 = 3 + (5 - 1)⋅3 = 15
Example (Infinite Sequence): {1,4,7,10,13,…}
a = 1 and d = 3
an = 1 + (n - 1)⋅3 = 3n - 2
Formula: Sum of Finite Arithmetic Sequence
Sn = n2{2a + (n - 1)d}
Example: {3,6,9,12,15}
a=3, d=3 and n=5
S5 = 52{2⋅3 + (5 - 1)⋅3} = 52 ✕ 18 = 45
Formula: Geometric Sequence
S5 = 52{2⋅3 + (5 - 1)⋅3} = 52 ✕ 18 = 45
Formula: Geometric Sequence
In a multiplicative sequence, each subsequent element is obtained by multiplying the previous element by a fixed number (called the ratio r).
Find the nth term;
an = a⋅rn-1
where,
an = n-th term
a = First Term
r = common ratio
n = number of terms
Example (Finite Sequence): {2,6,18,54}
a = 2 and r = 3
a4 = 2⋅3(4-1) = 54
Example (Infinite Sequence): {5,10,20,40,80,…}
a = 5 and r = 2
an = 5⋅2n-1
Formula: Sum of Finite Geometric Sequence
Sn = a 1 - rn1 - r (যদি r≠1)
Example: {2,6,18,54}
an = n-th term
a = First Term
r = common ratio
n = number of terms
Example (Finite Sequence): {2,6,18,54}
a = 2 and r = 3
a4 = 2⋅3(4-1) = 54
Example (Infinite Sequence): {5,10,20,40,80,…}
a = 5 and r = 2
an = 5⋅2n-1
Formula: Sum of Finite Geometric Sequence
Sn = a 1 - rn1 - r (যদি r≠1)
Example: {2,6,18,54}
a = 2, r = 3 and n = 4
S4 = 2 1 - 341 - 3= 2 1 - 81 -2 = 2 ✕ -40 -2 = 80
Formula: Sum of Infinite Geometric Sequence
If ∣r∣＜1, then the sum of infinite multiplicative sequences:
S4 = 2 1 - 341 - 3= 2 1 - 81 -2 = 2 ✕ -40 -2 = 80
Formula: Sum of Infinite Geometric Sequence
If ∣r∣＜1, then the sum of infinite multiplicative sequences:
S = a1 - r
Example: {1,21,41,81,…}
Example: {1,21,41,81,…}
a = 1 and r = 1 2
S = 1 ÷ (1 - 1 2) = 1 ✕ 2 = 2
Fibonacci Sequence
The Fibonacci sequence is a specific sequence where each term is the sum of the two preceding ones, usually starting with 0 and 1. The sequence looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
Series
A series is the sum of the terms of a sequence. If you have a sequence of numbers, the series is formed by adding these numbers together.
Finite Series
A finite series is the sum of the terms of a finite sequence. For example, if you have the sequence 2, 4, 6, 8, the corresponding finite series is 2 + 4 + 6 + 8 = 20.
Infinite Series
An infinite series is the sum of the terms of an infinite sequence. For example, the sum of the sequence 1+2+3+4+…. is an infinite series. In some cases, such series can converge to a finite value (like in the case of a geometric series with a common ratio between -1 and 1).
Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence. For example, if you have the arithmetic sequence 2, 5, 8, 11, the arithmetic series is 2 + 5 + 8 + 11.
Geometric Series
A geometric series is the sum of the terms of a geometric sequence. For example, if you have the geometric sequence 3, 6, 12, 24, the geometric series is 3 + 6 + 12 + 24.
Formula: Arithmetic Series
If there are n terms, then the nth term will be:
an = a + (n - 1)d
Sum of Arithmetic Series
To calculate the sum Sn of the first n terms of the parallelogram we can use the following formula:
Sn = n2{2a + (n - 1)d}
It can also be written differently:
It can also be written differently:
Sn = n2(a + l)
where, l is the nth term or last term.
Formula: Geometric Series
Let the first term of a geometric series be a and the common ratio r. Then the terms of the series will be: a,ar,ar2,ar3,….
Formula: Geometric Series
Let the first term of a geometric series be a and the common ratio r. Then the terms of the series will be: a,ar,ar2,ar3,….
If there are n terms, the nth term will be:
an = arn-1
The formula for the sum of the first n terms of the geometric series Sn is:
The formula for the sum of the first n terms of the geometric series Sn is:
Sn = a 1 - rn1 - r
where, a is the first term, r is the common ratio, and n is the number of terms.
Sum of Infinite Geometric Series
If the common ratio of the geometric series is ∣r∣＜1, then the sum of infinite terms of the series S is formula:
If the common ratio of the geometric series is ∣r∣＜1, then the sum of infinite terms of the series S is formula:
S = a1 - r
Example-
Let's say a geometric series starts from 2 and the common ratio is 3: 2,6,18,54,….
Sum of first five terms:
Example-
Let's say a geometric series starts from 2 and the common ratio is 3: 2,6,18,54,….
Sum of first five terms:
S5 = 2 1 - 351 - 3 = 2 1 - 243 -2 = 242
If the sum of an infinite number of terms is ∣r∣＜1 Let the common ratio, r = 1 2 এবং প্রথম পদ a=1:
If the sum of an infinite number of terms is ∣r∣＜1 Let the common ratio, r = 1 2 এবং প্রথম পদ a=1:
S = 1 ÷ (1 - 12) = 1 ÷ 0.5 = 2
Common Difference
The difference between two consecutive terms of a arithmetic Series is called the common difference. For example, in the series 2,5,8 the common Difference, d = 3
Common Ratio
The ratio between two consecutive terms of a geometric series is called a common ratio. For example, the common ratio in the sequence 3,9,27 is r = 3
Common Difference
The difference between two consecutive terms of a arithmetic Series is called the common difference. For example, in the series 2,5,8 the common Difference, d = 3
Common Ratio
The ratio between two consecutive terms of a geometric series is called a common ratio. For example, the common ratio in the sequence 3,9,27 is r = 3
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