Progression can be defined as a list of numbers having a particular pattern. The next number in a progression can be calculated using a formula. The relation between numbers must always be the same and relevant.
Progressions are mainly used to identify the nth term in the sequence. Based on the type of relation between numbers progressions are divided into three types.
Key difference between arithmetic and geometric progression are:
| Feature | Arithmetic Progression (AP) | Geometric Progression (GP) |
|---|---|---|
| Definition | A sequence of numbers in which each term is obtained by adding a constant to the previous term. | A sequence of numbers in which each term is obtained by multiplying the previous term by a constant. |
| General Form | a, a + d, a + 2d, . . . | a, ar, ar2, . . . |
| Common Difference/Ratio | Common difference (d) between consecutive terms is constant. | Common ratio (r) between consecutive terms is constant. |
| Formula for nth Term | an = a + (n − 1)d | an = a ⋅ rn−1 |
| Sum of n Terms | Sn = (n/2)[2a + (n − 1)d] | Sn = a(rn - 1)/(r - 1) [Where r ≠ 1] |
| Growth Pattern | Linear (terms increase or decrease linearly) | Exponential (terms increase or decrease exponentially) |
| Examples | 2, 4, 6, 8, . . . (common difference = 2) | 2, 6, 18, 54,. . . (common ratio = 3) |
| Applications | Used in financial calculations, such as EMIs, salaries, etc. | Used in compound interest, population growth, etc. |
Read More,
Sample Questions on Arithmetic and Geometric Progression
Question 1: Check if the sequence 5, 10, 15, 20, 25, 30,...is in Arithmetic Progression or not?
Solution:
a = 5
To check the given sequence is in Arithmetic Progression the sequence must follow condition,
a2- a1 = a3-a2
10-5 = 15-10
5 = 5
Since, the condition is satisfied the above sequence is in Arithmetic Progression.
Question 2: Calculate 20th term from the Arithmetic Progression 2, 6, 10, 14, 18, 22,....?
Solution:
a = 2
common difference (d) = a2 - a1 = 6 - 2 = 4
Since, nth term in the progression is given by formula
an = a + (n-1)d
here n = 20
a20 = 2 + (20-1)(4)
a20 = 2 + (19)(4)
a20 = 78
Therefore 20th term in the progression is 78.
Question 3: Calculate 10th term in the given Geometric Progression 2, 8, 32, 128,....?
Solution:
a = 2
Common ratio (r) = a2/a1 = 8/2 = 4
Since, nth term in Geometric Progression is given by formula
an = arn-1
Here n = 10
a10 = 2× (4)10-1
a10 = 524288
Therefore 10th term in the progression is 524288.
Question 4: Check if the sequence 5, 20, 80, 320....is in Geometric Progression or not?
Solution:
a = 5
To check the given sequence is in Geometric Progression the sequence must follow condition,
a2/a1 = a3/a2
20/5 = 80/20
4 = 4
Since, the condition is satisfied the above sequence is in Geometric Progression.
Question 5: Calculate the sum of the first 10 terms in the Arithmetic Progression 2, 6, 10, 14, 18, 22,...?
Solution:
a = 2
common difference (d) = a2 - a1 = 6 - 2 = 4
Since, sum of first n terms in a progression is given by formula
Sn = (n/2) (2a + (n-1)d)
Here n = 10
S10 = (10/2)(2*2 + (10-1)*4)
S10 = (5)(4 + 36)
S10 = 200
Therefore sum of first 10 terms in the given progression is 200
Question 6: Calculate the sum of the first 5 terms in the Geometric Progression 128, 32, 8, 2,...........?
Solution:
a = 128
Common ratio (r) = a2/a1 = 32/128 = 1/4 = 0.25
Since the sum of first n terms in an Infinite Geometric Progression is given by the formula,
Sn = a/(1-r), when |r|<1
Here n = 5
S5 = 128/(1-0.25)
S5 = 170.6
Therefore the sum of the first 5 terms in the given progression is 170.6
Question 7: Calculate the sum of the first 7 terms in the Geometric Progression 5, 20, 80, 320...20480?
Solution:
a = 5
Common ratio (r) = a2/a1 = 20/5 = 4
Since, sum of first n terms in an Finite Geometric Progression is given by formula
Sn = a(rn-1)/(r-1) r>1
Here n = 7
S7 = (5) (47-1)/(4-1)
S7 = 27305
Therefore sum of first 7 terms in the given progression is 27305