Progression:
When there is a definite relationship between the two successive terms of a series, we call the series as progressing series. The relation between the two successive terms decides the nature of the progression.
When the successive term is obtained on multiplying a given term with a constant number, the sequence so obtained is called geometric sequence. Example: 2, 4, 8, 16…
Convention:
Common ratio: The constant term that is multiplied with a given term of the sequence to obtain the successive term is called common ratio.
Sum of a geometric progression:
To find the sum of a geometric progression we adopt the following steps:
Step 1: Assume the sum of the given geometric progression as ‘S’.
Step 2: Find the common ratio of the progression.
Step 3: Multiply every term of the progression with ‘r’ to obtain S x r.
Step 4: Find the value of (Sr – S)
Step 5: Solve for S.
Example: Find the sum of 2 + 4 + 8 + 16 …256
Step 1: S = 2 + 4 + 8 + 16 …256
Step 2: Common ratio is 2
Step 3: 2S = 4 + 8 + 16 …512
Step 4: 2S – S = 512 -2
Step 5: S= 510.
Sigma Notation Series
If the first term and the common ratio of the geometric progression are represented by ‘a’ and ‘r’ respectively, then the geometric progression can be shown as
a + ar + ar^2 + ar^3… ar^(n-1)
It is a common practice in mathematics to represent the summation using sigma notation. The corresponding sigma notation for the above geometric summation is
Observe that, on substituting the values of k varying from zero to (n-1), we get the requisite geometric summation.
Special Series
Series can be obtained by finding the successive terms following a certain mathematical expression. Such series are called special series. The mathematical expression that is followed decides the algorithm to find the summation of the series.
Case 1: When the modulus value of the common ratio of the geometric sequence is less than ‘1’, the sequence will converge.
Example: 1 + 0.5 +0.25 …
Case 2: When the sequence is a rational function of ‘n’ with the modulus value less than ‘1’ for any value of n, the sum of the sequence will converge.
Example:
Case 3: When the denominator of the sequence is linear function of ‘n’, the summation of the sequence can be a convergent or divergent one.
Sum of Geometric progression:
By following the algorithm that is given below, we obtain the general formula for the summation of a geometrical sequence whose first terms is ’a’ and common ratio is ‘r’.
Step 1: Assume the sum of the given geometric progression as ‘S’.
Step 2: Find the common ratio of the progression.
Step 3: Multiply every term of the progression with ‘r’ to obtain S x r.
Step 4: Find the value of (Sr – S)
Step 5: Solve for S.
Nth term formula:
By applying the iterative method we obtain the nth term of a geometric sequence whose first terms is ‘a’ and common ratio is ‘r’ as an = a(r) (n-1).
Geometric progression examples:
Find the 15 term of the sequence 3, 6, 12…
First term = a = 3
Common ratio = r = 2.
15th term of the sequence is ar14 = 3(2)14 = 49152.
When there is a definite relationship between the two successive terms of a series, we call the series as progressing series. The relation between the two successive terms decides the nature of the progression.
When the successive term is obtained on multiplying a given term with a constant number, the sequence so obtained is called geometric sequence. Example: 2, 4, 8, 16…
Convention:
Common ratio: The constant term that is multiplied with a given term of the sequence to obtain the successive term is called common ratio.
Sum of a geometric progression:
To find the sum of a geometric progression we adopt the following steps:
Step 1: Assume the sum of the given geometric progression as ‘S’.
Step 2: Find the common ratio of the progression.
Step 3: Multiply every term of the progression with ‘r’ to obtain S x r.
Step 4: Find the value of (Sr – S)
Step 5: Solve for S.
Example: Find the sum of 2 + 4 + 8 + 16 …256
Step 1: S = 2 + 4 + 8 + 16 …256
Step 2: Common ratio is 2
Step 3: 2S = 4 + 8 + 16 …512
Step 4: 2S – S = 512 -2
Step 5: S= 510.
Sigma Notation Series
If the first term and the common ratio of the geometric progression are represented by ‘a’ and ‘r’ respectively, then the geometric progression can be shown as
a + ar + ar^2 + ar^3… ar^(n-1)
It is a common practice in mathematics to represent the summation using sigma notation. The corresponding sigma notation for the above geometric summation is
Observe that, on substituting the values of k varying from zero to (n-1), we get the requisite geometric summation.
Special Series
Series can be obtained by finding the successive terms following a certain mathematical expression. Such series are called special series. The mathematical expression that is followed decides the algorithm to find the summation of the series.
Case 1: When the modulus value of the common ratio of the geometric sequence is less than ‘1’, the sequence will converge.
Example: 1 + 0.5 +0.25 …
Case 2: When the sequence is a rational function of ‘n’ with the modulus value less than ‘1’ for any value of n, the sum of the sequence will converge.
Example:
Case 3: When the denominator of the sequence is linear function of ‘n’, the summation of the sequence can be a convergent or divergent one.
Sum of Geometric progression:
By following the algorithm that is given below, we obtain the general formula for the summation of a geometrical sequence whose first terms is ’a’ and common ratio is ‘r’.
Step 1: Assume the sum of the given geometric progression as ‘S’.
Step 2: Find the common ratio of the progression.
Step 3: Multiply every term of the progression with ‘r’ to obtain S x r.
Step 4: Find the value of (Sr – S)
Step 5: Solve for S.
Nth term formula:
By applying the iterative method we obtain the nth term of a geometric sequence whose first terms is ‘a’ and common ratio is ‘r’ as an = a(r) (n-1).
Geometric progression examples:
Find the 15 term of the sequence 3, 6, 12…
First term = a = 3
Common ratio = r = 2.
15th term of the sequence is ar14 = 3(2)14 = 49152.
No comments:
Post a Comment